# Reproducing kernel Hilbert $C^{*}$-module and kernel mean embeddings

Yuka Hashimoto<EMAIL_ADDRESS>
NTT Network Service Systems Laboratories, NTT Corporation
3-9-11, Midori-cho, Musashinoshi, Tokyo, 180-8585, Japan /
Graduate School of Science and Technology, Keio University
3-14-1, Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan Isao Ishikawa
<EMAIL_ADDRESS>
Center for Data Science, Ehime University
2-5, Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan /
Center for Advanced Intelligence Project, RIKEN
1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan Masahiro Ikeda
<EMAIL_ADDRESS>
Center for Advanced Intelligence Project, RIKEN
1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan /
Faculty of Science and Technology, Keio University
3-14-1, Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan Fuyuta Komura
<EMAIL_ADDRESS>
Faculty of Science and Technology, Keio University
3-14-1, Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan /
Center for Advanced Intelligence Project, RIKEN
1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan Takeshi Katsura
<EMAIL_ADDRESS>
Faculty of Science and Technology, Keio University
3-14-1, Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan /
Center for Advanced Intelligence Project, RIKEN
1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan Yoshinobu Kawahara
<EMAIL_ADDRESS>
Institute of Mathematics for Industry, Kyushu University
744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan /
Center for Advanced Intelligence Project, RIKEN
1-4-1, Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan

###### Abstract

Kernel methods have been among the most popular techniques in machine
learning, where learning tasks are solved using the property of reproducing
kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis
framework with reproducing kernel Hilbert $C^{*}$-module (RKHM) and kernel
mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS
or vector-valued RKHS (vvRKHS), analysis with RKHM enables us to capture and
extract structural properties in such as functional data. We show a branch of
theories for RKHM to apply to data analysis, including the representer
theorem, and the injectivity and universality of the proposed KME. We also
show RKHM generalizes RKHS and vvRKHS. Then, we provide concrete procedures
for employing RKHM and the proposed KME to data analysis.

Keywords: reproducing kernel Hilbert $C^{*}$-module, kernel mean embedding,
structured data, kernel PCA, interaction effects

##  1 Introduction

Kernel methods have been among the most popular techniques in machine learning
(Schölkopf and Smola, 2001), where learning tasks are solved using the
property of reproducing kernel Hilbert space (RKHS). RKHS is the space of
complex-valued functions equipped with an inner product determined by a
positive-definite kernel. One of the important tools with RKHS is kernel mean
embedding (KME). In KME, a probability distribution (or measure) is embedded
as a function in an RKHS (Smola et al., 2007; Muandet et al., 2017;
Sriperumbudur et al., 2011), which enables us to analyze distributions in
RKHSs.

Whereas much of the classical literature on RKHS approaches has focused on
complex-valued functions, RKHSs of vector-valued functions, i.e., vector-
valued RKHSs (vvRKHSs), have also been proposed (Micchelli and Pontil, 2005;
Álvarez et al., 2012; Lim et al., 2015; Minh et al., 2016; Kadri et al.,
2016). This allows us to learn vector-valued functions rather than complex-
valued functions.

In this paper, we develop a branch of theories on reproducing kernel Hilbert
$C^{*}$-module (RKHM) and propose a generic framework for data analysis with
RKHM. RKHM is a generalization of RKHS and vvRKHS in terms of $C^{*}$-algebra,
and we show that RKHM is a powerful tool to analyze structural properties in
such as functional data. An RKHM is constructed by a $C^{*}$-algebra-valued
positive definite kernel and characterized by a $C^{*}$-algebra-valued inner
product (see Definition 2.21). The theory of $C^{*}$-algebra has been
discussed in mathematics, especially in operator algebra theory. An important
example of $C^{*}$-algebra is $L^{\infty}(\Omega)$, where $\Omega$ is a
compact measure space. Another important example is
$\mathcal{B}(\mathcal{W})$, which denotes the space of bounded linear
operators on a Hilbert space $\mathcal{W}$. Note that
$\mathcal{B}(\mathcal{W})$ coincides with the space of matrices
$\mathbb{C}^{m\times m}$ if the Hilbert space $\mathcal{W}$ is finite
dimensional.

Although there are several advantages for studying RKHM compared with RKHS and
vvRKHS, those can be summarized into two points as follows: First, an RKHM is
a “Hilbert $C^{*}$-module”, which is mathematically more general than a
“Hilbert space”. The inner product in an RKHM is $C^{*}$-algebra-valued, which
captures more information than the complex-valued one in an RKHS or vvRKHS and
enables us to extract richer information. For example, if we set
$L^{\infty}(\Omega)$ as a $C^{*}$-algebra, we can control and extract features
of functional data such as derivatives, total variation, and frequency
components. Also, if we set $\mathcal{B}(\mathcal{W})$ as a $C^{*}$-algebra
and the inner product is described by integral operators, we can control and
extract features of continuous relationships between pairs of functional data.
This cannot be achieved, in principle, by RKHSs and vv-RKHSs. This is because
their inner products are complex-valued, where such information degenerates
into one complex value or is lost by discretizations of function into complex
values. Therefore, we cannot reconstruct the information from a vector in an
RKHS or vvRKHS. Second, RKHM generalizes RKHS and vvRKHS, that is, it can be
shown that we can reconstruct RKHSs and vvRKHSs from RKHMs. This implies that
existing algorithms with RKHSs and vvRKHSs are reconstructed by using the
framework of RKHM.

The theory of RKHM has been studied in mathematical physics and pure
mathematics (Itoh, 1990; Heo, 2008; Szafraniec, 2010). On the other hand, to
the best of our knowledge, as for the application of RKHM to data analysis, we
can find the only literature by Ye (2017), where only the case of setting the
space of matrices as a $C^{*}$-algebra is discussed. In this paper, we develop
a branch of theories on RKHM and propose a generic framework for data analysis
with RKHM. We show a theoretical property on minimization with respect to
orthogonal projections and give a representer theorem in RKHMs. These
properties are fundamental for data analysis that have been investigated and
applied in the cases of RKHS and vvRKHS, which has made RKHS and vvRKHS
widely-accepted tools for data analysis (Schölkopf et al., 2001). Moreover, we
define a KME in an RKHM, and provide theoretical results about the injectivity
of the proposed KME and the connection with universality of RKHM. Note that,
as is well known for RKHSs, these two properties have been actively studied to
theoretically guarantee the validity of kernel-based algorithms (Steinwart,
2001; Gretton et al., 2006; Fukumizu et al., 2007; Sriperumbudur et al.,
2011). Then, we apply the developed theories to generalize kernel PCA
(Schölkopf and Smola, 2001), analyze time-series data with the theory of
dynamical system, and analyze interaction effects for infinite dimensional
data.

The remainder of this paper is organized as follows. First, in Section 2, we
briefly review RKHS, vvRKHS, and the definition of RKHM. In Section 3, we
provide an overview of the motivation of studying RKHM for data analysis. In
Section 4, we show general properties of RKHM for data analysis and the
connection of RKHMs with RKHSs and vvRKHSs. In Sections 5, we propose a KME in
RKHMs, and show the connection between the injectivity of the KME and the
universality of RKHM. Then, in Section 6, we discuss applications of the
developed results to kernel PCA, time-series data analysis, and the analysis
of interaction effects in finite or infinite dimensional data. Finally, in
Section 7, we discuss the connection of RKHMs and the proposed KME with the
existing notions, and conclude the paper in Section 8.

##### Notations

Lowercase letters denote $\mathcal{A}$-valued coefficients (often by
$a,b,c,d$), vectors in a Hilbert $C^{*}$-module $\mathcal{M}$ (often by
$p,q,u,v$), or vectors in a Hilbert space $\mathcal{W}$ (often by $w,h$).
Lowercase Greek letters denote measures (often by $\mu,\nu,\lambda$) or
complex-valued coefficients (often by $\alpha,\beta$). Calligraphic capital
letters denote sets. And, bold lowercase letters denote vectors in
$\mathcal{A}^{n}$ for $n\in\mathbb{N}$ (a finite dimensional Hilbert
$C^{*}$-module). Also, we use $\sim$ for objects related to RKHSs. Moreover,
an inner product, an absolute value, and a norm in a space or a module
$\mathcal{S}$ (see Definitions 2.12 and 2.13) are denoted as
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{S}}$, $|\cdot|_{\mathcal{S}}$,
and $\|\cdot\|_{\mathcal{S}}$, respectively.

The typical notations in this paper are listed in Table 1.

Table 1: Notation table $\mathcal{A}$ |  A $C^{*}$-algebra
---|---
$1_{\mathcal{A}}$ |  The multiplicative identity in $\mathcal{A}$
$\mathcal{A}_{+}$ |  The subset of $\mathcal{A}$ composed of all positive elements in $\mathcal{A}$
$\leq_{\mathcal{A}}$ |  For $c,d\in\mathcal{A}$, $c\leq_{\mathcal{A}}d$ means $d-c$ is positive.
$<_{\mathcal{A}}$ |  For $c,d\in\mathcal{A}$, $c<d$ means $d-c$ is strictly positive, i.e., $d-c$ is positive and invertible.
$L^{\infty}(\Omega)$ |  The space of complex-valued $L^{\infty}$ functions on a measure space $\Omega$
$\mathcal{B}(\mathcal{W})$ |  The space of bounded linear operators on a Hilbert space $\mathcal{W}$
$\mathbb{C}^{m\times m}$ |  A set of all complex-valued $m\times m$ matrix
$\mathcal{M}$ |  A Hilbert $\mathcal{A}$-module
$\mathcal{X}$ |  A nonempty set for data
$C(\mathcal{X},\mathcal{Y})$ |  The space of $\mathcal{Y}$-valued continuous functions on $\mathcal{X}$ for topological spaces $\mathcal{X}$ and $\mathcal{Y}$
$n$ |  A natural number that represents the number of samples
$k$ |  An $\mathcal{A}$-valued positive definite kernel
$\phi$ |  The feature map endowed with $k$
$\mathcal{M}_{k}$ |  The RKHM associated with $k$
$\mathcal{S}^{\mathcal{X}}$ |  The set of all functions from a set $\mathcal{X}$ to a space $\mathcal{S}$
$\tilde{k}$ |  A complex-valued positive definite kernel
$\tilde{\phi}$ |  The feature map endowed with $\tilde{k}$
$\mathcal{H}_{\tilde{k}}$ |  The RKHS associated with $\tilde{k}$
$\mathcal{H}_{k}^{\operatorname{v}}$ |  The vvRKHS associated with $k$
$\mathcal{D}(\mathcal{X},\mathcal{A})$ |  The set of all $\mathcal{A}$-valued finite regular Borel measures
$\Phi$ |  The proposed KME in an RKHM
$\delta_{x}$ |  The $\mathcal{A}$-valued Dirac measure defined as $\delta_{x}(E)=1_{\mathcal{A}}$ for $x\in E$ and $\delta_{x}(E)=0$ for $x\notin E$
$\tilde{\delta}_{x}$ |  The complex-valued Dirac measure defined as $\tilde{\delta}_{x}(E)=1$ for $x\in E$ and $\tilde{\delta}_{x}(E)=0$ for $x\notin E$
$\chi_{E}$ |  The indicator function of a Borel set $E$ on $\mathcal{X}$
${C}_{0}(\mathcal{X},\mathcal{A})$ |  The space of all continuous $\mathcal{A}$-valued functions on $\mathcal{X}$ vanishing at infinity
$\mathbf{G}$ |  The $\mathcal{A}$-valued Gram matrix defined as $\mathbf{G}_{i,j}=k(x_{i},x_{j})$ for given samples $x_{1},\ldots,x_{n}\in\mathcal{X}$
$p_{j}$ |  The $j$-th principal axis generated by kernel PCA with an RKHM
$r$ |  A natural number that represents the number of principal axes
$Df_{\mathbf{c}}$ |  The Gâteaux derivative of a function $f:\mathcal{M}\to\mathcal{A}$ at $\mathbf{c}\in\mathcal{M}$
$\nabla f_{\mathbf{c}}$ |  The gradient of a function $f:\mathcal{M}\to\mathcal{A}$ at $\mathbf{c}\in\mathcal{M}$

##  2 Background

We briefly review RKHS and vvRKHS in Subsections 2.1 and 2.2, respectively.
Then, we review $C^{*}$-algebra and $C^{*}$-module in Subsection 2.3, Hilbert
$C^{*}$-module in Subsection 2.4, and RKHM in Subsection 2.5.

###  2.1 Reproducing kernel Hilbert space (RKHS)

We review the theory of RKHS. An RKHS is a Hilbert space and useful for
extracting nonlinearity or higher-order moments of data (Schölkopf and Smola,
2001; Saitoh and Sawano, 2016).

We begin by introducing positive definite kernels. Let $\mathcal{X}$ be a non-
empty set for data, and $\tilde{k}$ be a positive definite kernel, which is
defined as follows:

######  Definition 2.1 (Positive definite kernel)

A map $\tilde{k}:\mathcal{X}\times\mathcal{X}\to\mathbb{C}$ is called a
positive definite kernel if it satisfies the following conditions:

  1. 1.

$\tilde{k}(x,y)=\overline{\tilde{k}(y,x)}$ for $x,y\in\mathcal{X}$,

  2. 2.

$\sum_{i,j=1}^{n}\overline{\alpha}_{i}\alpha_{j}\tilde{k}(x_{i},x_{j})\geq 0$
for $n\in\mathbb{N}$, $\alpha_{i}\in\mathbb{C}$, $x_{i}\in\mathcal{X}$.

Let $\tilde{\phi}:\mathcal{X}\to\mathbb{C}^{\mathcal{X}}$ be a map defined as
$\tilde{\phi}(x)=\tilde{k}(\cdot,x)$. With $\tilde{\phi}$, the following space
as a subset of $\mathbb{C}^{\mathcal{X}}$ is constructed:

$\mathcal{H}_{\tilde{k},0}:=\bigg{\\{}\sum_{i=1}^{n}\alpha_{i}\tilde{\phi}(x_{i})\bigg{|}\
n\in\mathbb{N},\ \alpha_{i}\in\mathbb{C},\ x_{i}\in\mathcal{X}\bigg{\\}}.$

Then, a map
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{\tilde{k}}}:\mathcal{H}_{\tilde{k},0}\times\mathcal{H}_{\tilde{k},0}\to\mathbb{C}$
is defined as follows:

$\bigg{\langle}\sum_{i=1}^{n}\alpha_{i}\tilde{\phi}(x_{i}),\sum_{j=1}^{l}\beta_{j}\tilde{\phi}(y_{j})\bigg{\rangle}_{\mathcal{H}_{\tilde{k}}}:=\sum_{i=1}^{n}\sum_{j=1}^{l}\overline{\alpha_{i}}\beta_{j}\tilde{k}(x_{i},y_{j}).$

By the properties in Definition 2.1 of $\tilde{k}$,
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{\tilde{k}}}$ is well-
defined, satisfies the axiom of inner products, and has the reproducing
property, that is,

$\langle\tilde{\phi}(x),v\rangle_{\mathcal{H}_{\tilde{k}}}=v(x)$

for $v\in\mathcal{H}_{\tilde{k},0}$ and $x\in\mathcal{X}$.

The completion of $\mathcal{H}_{\tilde{k},0}$ is called the RKHS associated
with $\tilde{k}$ and denoted as $\mathcal{H}_{\tilde{k}}$. It can be shown
that $\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{\tilde{k}}}$ is
extended continuously to $\mathcal{H}_{\tilde{k}}$ and the map
$\mathcal{H}_{\tilde{k}}\ni
v\mapsto(x\mapsto\langle\tilde{\phi}(x),v\rangle_{\mathcal{H}_{\tilde{k}}})\in\mathbb{C}^{\mathcal{X}}$
is injective. Thus, $\mathcal{H}_{\tilde{k}}$ is regarded to be a subset of
$\mathbb{C}^{\mathcal{X}}$ and has the reproducing property. Also,
$\mathcal{H}_{\tilde{k}}$ is determined uniquely.

The map $\tilde{\phi}$ maps data into $\mathcal{H}_{\tilde{k}}$ and is called
the feature map. Since the dimension of $\mathcal{H}_{\tilde{k}}$ is higher
(often infinite dimensional) than that of $\mathcal{X}$, complicated behaviors
of data in $\mathcal{X}$ are expected to be transformed into simple ones in
$\mathcal{H}_{\tilde{k}}$ (Schölkopf and Smola, 2001).

###  2.2 Vector-valued RKHS (vvRKHS)

We review the theory of vvRKHS. Complex-valued functions in RKHSs are
generalized to vector-valued functions in vvRKHSs. Similar to the case of
RKHS, we begin by introducing positive definite kernels. Let $\mathcal{X}$ be
a non-empty set for data and $\mathcal{W}$ be a Hilbert space. In addition,
let $k$ be an operator-valued positive definite kernel, which is defined as
follows:

######  Definition 2.2 (Operator-valued positive definite kernel)

A map $k:\mathcal{X}\times\mathcal{X}\to\mathcal{B}(\mathcal{W})$ is called an
operator-valued positive definite kernel if it satisfies the following
conditions:

  1. 1.

$k(x,y)=k(y,x)^{*}$ for $x,y\in\mathcal{X}$,

  2. 2.

$\sum_{i,j=1}^{n}\left\langle
w_{i},k(x_{i},x_{j})w_{j}\right\rangle_{\mathcal{W}}\geq 0$ for
$n\in\mathbb{N}$, $w_{i}\in\mathcal{W}$, $x_{i}\in\mathcal{X}$.

Here, ∗ represents the adjoint.

Let $\phi:\mathcal{X}\to\mathcal{B}(\mathcal{W})^{\mathcal{X}}$ be a map
defined as $\phi(x)=k(\cdot,x)$. With $\phi$, the following space as a subset
of $\mathcal{W}^{\mathcal{X}}$ is constructed:

$\mathcal{H}_{k,0}^{\operatorname{v}}:=\bigg{\\{}\sum_{i=1}^{n}\phi(x_{i})w_{i}\bigg{|}\
n\in\mathbb{N},\ w_{i}\in\mathcal{W},\ x_{i}\in\mathcal{X}\bigg{\\}}.$

Then, a map
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}:\mathcal{H}_{k,0}^{\operatorname{v}}\times\mathcal{H}_{k,0}^{\operatorname{v}}\to\mathbb{C}$
is defined as follows:

$\displaystyle\bigg{\langle}\sum_{i=1}^{n}\phi(x_{i})w_{i},\sum_{j=1}^{l}\phi(y_{j})h_{j}\bigg{\rangle}_{\mathcal{H}_{k}^{\operatorname{v}}}:=\sum_{i=1}^{n}\sum_{j=1}^{l}\left\langle
w_{i},k(x_{i},y_{j})h_{j}\right\rangle_{\mathcal{W}}.$

By the properties in Definition 2.2 of $k$,
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}$ is
well-defined, satisfies the axiom of inner products, and has the reproducing
property, that is,

$\left\langle\phi(x)w,u\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}=\left\langle
w,u(x)\right\rangle_{\mathcal{W}}$ (1)

for $u\in\mathcal{H}_{k,0}^{{\operatorname{v}}}$, $x\in\mathcal{X}$, and
$w\in\mathcal{W}$.

The completion of $\mathcal{H}_{k,0}^{\operatorname{v}}$ is called the vvRKHS
associated with $k$ and denoted as $\mathcal{H}_{k}^{\operatorname{v}}$. Note
that since an inner product in $\mathcal{H}_{k}^{\operatorname{v}}$ is defined
with the complex-valued inner product in $\mathcal{W}$, it is complex-valued.

###  2.3 $C^{*}$-algebra and Hilbert $C^{*}$-module

A $C^{*}$-algebra and a $C^{*}$-module are generalizations of the space of
complex numbers $\mathbb{C}$ and a vector space, respectively. In this paper,
we denote a $C^{*}$-algebra by $\mathcal{A}$ and a $C^{*}$-module by
$\mathcal{M}$, respectively. As we see below, many complex-valued notions can
be generalized to $\mathcal{A}$-valued.

A $C^{*}$-algebra is defined as a Banach space equipped with a product
structure, an involution $(\cdot)^{*}:\mathcal{A}\rightarrow\mathcal{A}$, and
additional properties. We denote the norm of $\mathcal{A}$ by
$\|\cdot\|_{\mathcal{A}}$.

######  Definition 2.3 (Algebra)

A set $\mathcal{A}$ is called an algebra on a filed $\mathbb{F}$ if it is a
vector space equipped with an operation
$\cdot:\mathcal{A}\times\mathcal{A}\to\mathcal{A}$ which satisfies the
following conditions for $b,c,d\in\mathcal{A}$ and $\alpha\in\mathbb{F}$:

$\bullet$ $(b+c)\cdot
d={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}b}\cdot
d+c\cdot d$, $\bullet$ $b\cdot(c+d)=b\cdot c+b\cdot d$, $\bullet$ $(\alpha
c)\cdot d=\alpha(c\cdot d)=c\cdot(\alpha d)$.

The symbol $\cdot$ is omitted when it does not cause confusion.

######  Definition 2.4 ($C^{*}$-algebra)

A set $\mathcal{A}$ is called a $C^{*}$-algebra if it satisfies the following
conditions:

  1. 1.

$\mathcal{A}$ is an algebra over $\mathbb{C}$ and equipped with a bijection
$(\cdot)^{*}:\mathcal{A}\to\mathcal{A}$ that satisfies the following
conditions for $\alpha,\beta\in\mathbb{C}$ and $c,d\in\mathcal{A}$:

$\bullet$ $(\alpha c+\beta
d)^{*}=\overline{\alpha}c^{*}+\overline{\beta}d^{*}$, $\bullet$
$(cd)^{*}=d^{*}c^{*}$, $\bullet$ $(c^{*})^{*}=c$.

  2. 2.

$\mathcal{A}$ is a normed space with $\|\cdot\|_{\mathcal{A}}$, and for
$c,d\in\mathcal{A}$,
$\|cd\|_{\mathcal{A}}\leq\|c\|_{\mathcal{A}}\|d\|_{\mathcal{A}}$ holds. In
addition, $\mathcal{A}$ is complete with respect to $\|\cdot\|_{\mathcal{A}}$.

  3. 3.

For $c\in\mathcal{A}$, $\|c^{*}c\|_{\mathcal{A}}=\|c\|_{\mathcal{A}}^{2}$
holds.

######  Definition 2.5 (Multiplicative identity and unital $C^{*}$-algebra)

The multiplicative identity of $\mathcal{A}$ is the element $a\in\mathcal{A}$
which satisfies $ac=ca=c$ for any $c\in\mathcal{A}$. We denote by
$1_{\mathcal{A}}$ the multiplicative identity of $\mathcal{A}$. If a
$C^{*}$-algebra $\mathcal{A}$ has the multiplicative identity, then it is
called a unital $C^{*}$-algebra.

###### Example 2.6

Important examples of (unital) $C^{*}$-algebras are $L^{\infty}(\Omega)$ and
$\mathcal{B}(\mathcal{W})$, i.e., the space of complex-valued $L^{\infty}$
functions on a $\sigma$-finite measure space $\Omega$ and the space of bounded
linear operators on a Hilbert space $\mathcal{W}$, respectively.

  1. 1.

For $\mathcal{A}=L^{\infty}(\Omega)$, the product of two functions
$c,d\in\mathcal{A}$ is defined as $(cd)(t)=c(t)d(t)$ for any $t\in\Omega$, the
involution is defined as $c(t)=\overline{c(t)}$, the norm is the
$L^{\infty}$-norm, and the multiplicative identity is the constant function
whose value is $1$ at almost everywhere $t\in\Omega$.

  2. 2.

For $\mathcal{A}=\mathcal{B}(\mathcal{W})$, the product structure is the
product (the composition) of operators, the involution is the adjoint, the
norm $\|\cdot\|_{\mathcal{A}}$ is the operator norm, and the multiplicative
identity is the identity map.

In fact, by the Gelfand–Naimark theorem (see, for example, Murphy (1990)), any
$C^{*}$-algebra can be regarded as a subalgebra of $\mathcal{B}(\mathcal{W})$
for some Hilbert space $\mathcal{W}$. Therefore, considering the case of
$\mathcal{A}=\mathcal{B}(\mathcal{W})$ is sufficient for applications.

The positiveness is also important in $C^{*}$-algebras.

######  Definition 2.7 (Positive)

An element $c$ of $\mathcal{A}$ is called positive if there exists
$d\in\mathcal{A}$ such that $c=d^{*}d$ holds. For a unital $C^{*}$-algebra
$\mathcal{A}$, if a positive element $c\in\mathcal{A}$ is invertible, i.e.,
there exists $d\in\mathcal{A}$ such that $cd=dc=1_{\mathcal{A}}$, then $c$ is
called strictly positive. For $c,d\in\mathcal{A}$, we denote
$c\leq_{\mathcal{A}}d$ if $d-c$ is positive and $c<_{\mathcal{A}}d$ if $d-c$
is strictly positive. We denote by $\mathcal{A}_{+}$ the subset of
$\mathcal{A}$ composed of all positive elements in $\mathcal{A}$.

###### Example 2.8

  1. 1.

For $\mathcal{A}=L^{\infty}(\Omega)$, a function $c\in\mathcal{A}$ is positive
if and only if $c(t)\geq 0$ for almost everywhere $t\in\Omega$, and strictly
positive if and only if $c(t)>0$ for almost everywhere $t\in\Omega$.

  2. 2.

For $\mathcal{A}=\mathcal{B}(\mathcal{W})$, the positiveness is equivalent to
the positive semi-definiteness of self-adjoint operators and the strictly
positiveness is equivalent to the positive definiteness of self-adjoint
operators.

The positiveness provides us the (pre) order in $\mathcal{A}$ and, thus,
enables us to consider optimization problems in $\mathcal{A}$.

######  Definition 2.9 (Supremum and infimum)

  1. 1.

For a subset $\mathcal{S}$ of $\mathcal{A}$, $a\in\mathcal{A}$ is said to be
an upper bound with respect to the order $\leq_{\mathcal{A}}$, if
$d\leq_{\mathcal{A}}a$ for any $d\in\mathcal{S}$. Then, $c\in\mathcal{A}$ is
said to be a supremum of $\mathcal{S}$, if $c\leq_{\mathcal{A}}a$ for any
upper bound $a$ of $\mathcal{S}$.

  2. 2.

For a subset $\mathcal{S}$ of $\mathcal{A}$, $a\in\mathcal{A}$ is said to be a
lower bound with respect to the order $\leq_{\mathcal{A}}$, if
$a\leq_{\mathcal{A}}d$ for any $d\in\mathcal{S}$. Then, $c\in\mathcal{A}$ is
said to be a infimum of $\mathcal{S}$, if $a\leq_{\mathcal{A}}c$ for any lower
bound $a$ of $\mathcal{S}$.

We now introduce a $C^{*}$-module over $\mathcal{A}$, which is a
generalization of the vector space.

######  Definition 2.10 (Right multiplication)

Let $\mathcal{M}$ be an abelian group with operation $+$. For
$c,d\in\mathcal{A}$ and $u,v\in\mathcal{M}$, if an operation
$\cdot:\mathcal{M}\times\mathcal{A}\to\mathcal{M}$ satisfies

  1. 1.

$(u+v)\cdot c=u\cdot c+v\cdot c$,

  2. 2.

$u\cdot(c+d)=u\cdot c+u\cdot d$,

  3. 3.

$u\cdot(cd)=(u\cdot{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c})\cdot{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}d}$,

  4. 4.

$u\cdot 1_{\mathcal{A}}=u$ if $\mathcal{A}$ is unital,

then, $\cdot$ is called a (right) $\mathcal{A}$-multiplication. The
multiplication $u\cdot c$ is usually denoted as $uc$.

######  Definition 2.11 ($C^{*}$-module)

Let $\mathcal{M}$ be an abelian group with operation $+$. If $\mathcal{M}$ is
equipped with a (right) $\mathcal{A}$-multiplication, $\mathcal{M}$ is called
a (right) $C^{*}$-module over $\mathcal{A}$.

In this paper, we consider column vectors rather than row vectors for
representing $\mathcal{A}$-valued coefficients, and column vectors act on the
right. Therefore, we consider right multiplications. However, considering row
vectors and left multiplications instead of column vectors and right
multiplications is also possible.

###  2.4 Hilbert $C^{*}$-module

A Hilbert $C^{*}$-module is a generalization of a Hilbert space. We first
consider an $\mathcal{A}$-valued inner product, which is a generalization of a
complex-valued inner product, and then, introduce the definition of a Hilbert
$C^{*}$-module.

######  Definition 2.12 ($\mathcal{A}$-valued inner product)

A $\mathbb{C}$-linear map with respect to the second variable
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}}:\mathcal{M}\times\mathcal{M}\to\mathcal{A}$
is called an $\mathcal{A}$-valued inner product if it satisfies the following
properties for $u,v,p\in\mathcal{M}$ and $c,d\in\mathcal{A}$:

  1. 1.

$\left\langle u,vc+pd\right\rangle_{\mathcal{M}}=\left\langle
u,v\right\rangle_{\mathcal{M}}c+\left\langle u,p\right\rangle_{\mathcal{M}}d$,

  2. 2.

$\left\langle v,u\right\rangle_{\mathcal{M}}=\left\langle
u,v\right\rangle_{\mathcal{M}}^{*}$,

  3. 3.

$\left\langle u,u\right\rangle_{\mathcal{M}}\geq_{\mathcal{A}}0$,

  4. 4.

If $\left\langle u,u\right\rangle_{\mathcal{M}}=0$ then $u=0$.

######  Definition 2.13 ($\mathcal{A}$-valued absolute value and norm)

For $u\in\mathcal{M}$, the $\mathcal{A}$-valued absolute value
$|u|_{\mathcal{M}}$ on $\mathcal{M}$ is defined by the positive element
$|u|_{\mathcal{M}}$ of $\mathcal{A}$ such that
$|u|_{\mathcal{M}}^{2}=\left\langle u,u\right\rangle_{\mathcal{M}}$. The
(real-valued) norm $\|\cdot\|_{\mathcal{M}}$ on $\mathcal{M}$ is defined by
$\|u\|_{\mathcal{M}}=\big{\|}|u|_{\mathcal{M}}\big{\|}_{\mathcal{A}}$.

Since the absolute value $|\cdot|_{\mathcal{M}}$ takes values in
$\mathcal{A}$, it behaves more complicatedly. For example, the triangle
inequality does not hold for the absolute value. However, it provides us with
more information than the norm $\|\cdot\|_{\mathcal{M}}$ (which is real-
valued). For example, let $\mathcal{M}={\mathcal{A}=}\mathbb{C}^{m\times m}$,
$c=\operatorname{diag}\\{\alpha,0,\ldots,0\\}$, and
$d=\operatorname{diag}\\{\alpha,\ldots,\alpha\\}$, where
$\alpha\in\mathbb{C}$. Then, $\|c\|_{\mathcal{M}}=\|d\|_{\mathcal{M}}$, but
$|c|_{\mathcal{M}}\neq|d|_{\mathcal{M}}$. For a self-adjoint matrix, the
absolute value describes the whole spectrum of it, but the norm only describes
the largest eigenvalue.

######  Definition 2.14 (Hilbert $C^{*}$-module)

Let $\mathcal{M}$ be a (right) $C^{*}$-module over $\mathcal{A}$ equipped with
an $\mathcal{A}$-valued inner product defined in Definition 2.12. If
$\mathcal{M}$ is complete with respect to the norm $\|\cdot\|_{\mathcal{M}}$,
it is called a Hilbert $C^{*}$-module over $\mathcal{A}$ or Hilbert
$\mathcal{A}$-module.

###### Example 2.15

A simple example of Hilbert $C^{*}$ modules over $\mathcal{A}$ is
$\mathcal{A}^{n}$ for a natural number $n$. The $\mathcal{A}$-valued inner
product between $\mathbf{c}=[c_{1},\ldots,c_{n}]^{T}$ and
$\mathbf{d}=[d_{1},\ldots,d_{n}]^{T}$ is defined as
$\left\langle\mathbf{c},\mathbf{d}\right\rangle_{\mathcal{A}^{n}}=\sum_{i=1}^{n}c_{i}^{*}d_{i}$.
The absolute value and norm in $\mathcal{A}^{n}$ are given as
$|\mathbf{c}|_{\mathcal{A}^{n}}^{2}=(\sum_{i=1}^{n}c_{i}^{*}c_{i})$ and
$\|\mathbf{c}\|_{\mathcal{A}^{n}}=\|\sum_{i=1}^{n}c_{i}^{*}c_{i}\|_{\mathcal{A}}^{1/2}$,
respectively.

Similar to the case of Hilbert spaces, the following Cauchy–Schwarz inequality
for $\mathcal{A}$-valued inner products is available (Lance, 1995, Proposition
1.1).

######  Lemma 2.16 (Cauchy–Schwarz inequality)

For $u,v\in\mathcal{M}$, the following inequality holds:

$|\left\langle
u,v\right\rangle_{\mathcal{M}}|_{\mathcal{A}}^{2}\;\leq_{\mathcal{A}}\|u\|_{\mathcal{M}}^{2}\left\langle
v,v\right\rangle_{\mathcal{M}}.$

An important property associated with an inner product is the orthonormality.
The orthonormality plays an important role in data analysis. For example, an
orthonormal basis constructs orthogonal projections and an orthogonally
projected vector minimizes the deviation from its original vector in the
projected space. Therefore, we also introduce the orthonormality in Hilbert
$C^{*}$-module. See, for example, Definition 1.2 in (Bakić and Guljaš, 2001)
for more details.

######  Definition 2.17 (Normalized)

A vector $q\in\mathcal{M}$ is normalized if $0\neq\left\langle
q,q\right\rangle_{\mathcal{M}}=\left\langle
q,q\right\rangle_{\mathcal{M}}^{2}$.

Note that in the case of a general $C^{*}$-valued inner product, for a
normalized vector $q$, $\left\langle q,q\right\rangle_{\mathcal{M}}$ is not
always equal to the identity of $\mathcal{A}$ in contrast to the case of a
complex-valued inner product.

######  Definition 2.18 (Orthonormal system and basis)

Let $\mathcal{I}$ be an index set. A set
$\mathcal{S}=\\{q_{i}\\}_{i\in\mathcal{I}}\subseteq\mathcal{M}$ is called an
orthonormal system (ONS) of $\mathcal{M}$ if $q_{i}$ is normalized for any
$i\in\mathcal{I}$ and $\left\langle q_{i},q_{j}\right\rangle_{\mathcal{M}}=0$
for $i\neq j$. We call $\mathcal{S}$ an orthonormal basis (ONB) if the module
generated by $\mathcal{S}$ is an ONS and dense in $\mathcal{M}$.

In Hilbert $C^{*}$-modules, $\mathcal{A}$-linear is often used instead of
$\mathbb{C}$-linear.

######  Definition 2.19 ($\mathcal{A}$-linear operator)

Let $\mathcal{M}_{1},\mathcal{M}_{2}$ be Hilbert $\mathcal{A}$-modules. A
linear map $L:\mathcal{M}_{1}\to\mathcal{M}_{2}$ is referred to as
$\mathcal{A}$-linear if it satisfies $L(uc)=(Lu)c\;$ for any $u\in\mathcal{M}$
and $c\in\mathcal{A}$.

######  Definition 2.20 ($\mathcal{A}$-linearly independent)

The set $\mathcal{S}$ of $\mathcal{M}$ is said to be $\mathcal{A}$-linearly
independent if it satisfies the following condition: For any finite subset
$\\{v_{1},\ldots,v_{n}\\}$ of $\mathcal{S}$, if $\sum_{i=1}^{n}v_{i}c_{i}=0$
for $c_{i}\in\mathcal{A}$, then $c_{i}=0$ for $i=1,\ldots,n$.

For further details about $C^{*}$-algebra, $C^{*}$-module, and Hilbert
$C^{*}$-module, refer to Murphy (1990); Lance (1995).

###  2.5 Reproducing kernel Hilbert $C^{*}$-module (RKHM)

We summarize the theory of RKHM, which is discussed, for example, in Heo
(2008).

Similar to the case of RKHS, we begin by introducing an $\mathcal{A}$-valued
generalization of a positive definite kernel on a non-empty set $\mathcal{X}$
for data.

######  Definition 2.21 ($\mathcal{A}$-valued positive definite kernel)

An $\mathcal{A}$-valued map $k:\ \mathcal{X}\times\mathcal{X}\to\mathcal{A}$
is called a positive definite kernel if it satisfies the following conditions:

  1. 1.

$k(x,y)=k(y,x)^{*}$ for $x,y\in\mathcal{X}$,

  2. 2.

$\sum_{i,j=1}^{n}c_{i}^{*}k(x_{i},x_{j})c_{j}\geq_{\mathcal{A}}0$ for
$n\in\mathbb{N}$, $c_{i}\in\mathcal{A}$, $x_{i}\in\mathcal{X}$.

###### Example 2.22

  1. 1.

Let $\mathcal{X}=C([0,1]^{m})$. Let $\mathcal{A}=L^{\infty}([0,1])$ and let
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be defined as
$k(x,y)(t)=\int_{[0,1]^{m}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{(t-x(s))}}(t-y(s))ds$
for $t\in[0,1]$. Then, for $x_{1},\ldots,x_{n}\in\mathcal{X}$,
$c_{1},\ldots,c_{n}\in\mathcal{A}$ and $t\in[0,1]$, we have

$\displaystyle\sum_{i,j=1}^{n}c_{i}^{*}(t)k(x_{i},x_{j})(t)c_{j}(t)$
$\displaystyle=\int_{[0,1]^{m}}\sum_{i,j=1}^{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{c_{i}(t)(t-x_{i}(s))}}(t-x_{j}(s))c_{j}(t)ds$
$\displaystyle=\int_{[0,1]^{m}}\sum_{i=1}^{n}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{c_{i}(t)(t-x_{i}(s))}}\sum_{j=1}^{n}(t-x_{j}(s))c_{j}(t)ds\geq
0$

for $t\in[0,1]$. Thus, $k$ is an $\mathcal{A}$-valued positive definite
kernel.

  2. 2.

Let $\mathcal{A}=L^{\infty}([0,1])$ and
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be defined such that
$k(x,y)(t)$ is a complex-valued positive definite kernel for any $t\in[0,1]$.
Then, $k$ is an $\mathcal{A}$-valued positive definite kernel.

  3. 3.

Let $\mathcal{W}$ be a separable Hilbert space and let
$\\{e_{i}\\}_{i=1}^{\infty}$ be an orthonormal basis of $\mathcal{W}$. Let
$\mathcal{A}=\mathcal{B}(\mathcal{W})$. Let
$k_{i}:\mathcal{X}\times\mathcal{X}\to\mathbb{C}$ be a complex-valued positive
definite kernel for any $i=1,2,\ldots$. Assume for any $x\in\mathcal{X}$,
there exists $C>0$ such that for any $i=1,2,\ldots$, $|k_{i}(x,x)|\leq C$
holds. Let $k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be defined as
$k(x,y)e_{i}=k_{i}(x,y)e_{i}$. Then, for $x_{1},\ldots,x_{n}\in\mathcal{X}$,
$c_{1},\ldots,c_{n}\in\mathcal{A}$ and $w\in\mathcal{W}$, we have

$\displaystyle\bigg{\langle}w,\bigg{(}\sum_{i,j=1}^{n}c_{i}^{*}k(x_{i},x_{j})c_{j}\bigg{)}w\bigg{\rangle}_{\mathcal{W}}$
$\displaystyle=\sum_{i,j=1}^{n}\sum_{l=1}^{\infty}\left\langle\alpha_{i,l}e_{l},k(x_{i},x_{j})\alpha_{j,l}e_{l}\right\rangle_{\mathcal{W}}$
$\displaystyle=\sum_{l=1}^{\infty}\sum_{i,j=1}^{n}\overline{\alpha_{i,l}}\alpha_{j,l}\tilde{k}_{l}(x_{i},x_{j})\geq
0,$

where $c_{i}w=\sum_{l=1}^{\infty}\alpha_{i,l}e_{l}$ is the expansion with
respect to $\\{e_{i}\\}_{i=1}^{\infty}$. Thus, $k$ is an $\mathcal{A}$-valued
positive definite kernel.

  4. 4.

Let $\mathcal{X}=C(\Omega,\mathcal{Y})$ and $\mathcal{W}=L^{2}(\Omega)$ for a
topological space $\Omega$ with a finite Borel measure and a topological space
$\mathcal{Y}$. Let $\mathcal{A}=\mathcal{B}(\mathcal{W})$, and
$\tilde{k}:\mathcal{Y}\times\mathcal{Y}\to\mathbb{C}$ be a complex-valued
bounded continuous positive definite kernel. Moreover, let
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be defined as
$(k(x,y)w)(s)=\int_{t\in\Omega}\tilde{k}(x(s),y(t))w(t)dt$. Then, for
$x_{1},\ldots,x_{n}\in\mathcal{X}$, $c_{1},\ldots,c_{n}\in\mathcal{A}$ and
$w\in\mathcal{W}$, we have

$\displaystyle\bigg{\langle}w,\bigg{(}\sum_{i,j=1}^{n}c_{i}^{*}k(x_{i},x_{j})c_{j}\bigg{)}w\bigg{\rangle}_{\mathcal{W}}$
$\displaystyle=\int_{t\in\Omega}\int_{s\in\Omega}\sum_{i,j=1}^{n}\overline{d_{i}(s)}\tilde{k}(x_{i}(s),x_{j}(t))d_{j}(t)dsdt\geq
0,$

where $d_{i}=c_{i}w$. Thus, $k$ is an $\mathcal{A}$-valued positive definite
kernel.

Let $\phi:\mathcal{X}\to\mathcal{A}^{\mathcal{X}}$ be the feature map
associated with $k$, which is defined as $\phi(x)=k(\cdot,x)$ for
$x\in\mathcal{X}$. Similar to the case of RKHS, we construct the following
$C^{*}$-module composed of $\mathcal{A}$-valued functions by means of $\phi$:

$\mathcal{M}_{k,0}:=\bigg{\\{}\sum_{i=1}^{n}\phi(x_{i})c_{i}\bigg{|}\
n\in\mathbb{N},\ c_{i}\in\mathcal{A},\ x_{i}\in\mathcal{X}\bigg{\\}}.$

An $\mathcal{A}$-valued map
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}_{k}}:\mathcal{M}_{k,0}\times\mathcal{M}_{k,0}\to\mathcal{A}$
is defined as follows:

$\bigg{\langle}\sum_{i=1}^{n}\phi(x_{i})c_{i},\sum_{j=1}^{l}\phi(y_{j})d_{j}\bigg{\rangle}_{\mathcal{M}_{k}}:=\sum_{i=1}^{n}\sum_{j=1}^{l}c_{i}^{*}k(x_{i},y_{j})d_{j}.$

By the properties in Definition 2.21 of $k$,
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}_{k}}$ is well-defined and
has the reproducing property

$\left\langle\phi(x),v\right\rangle_{\mathcal{M}_{k}}=v(x)$

for $v\in\mathcal{M}_{k,0}$ and $x\in\mathcal{X}$. Also, it satisfies the
properties in Definition 2.12. As a result,
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}_{k}}$ is shown to be an
$\mathcal{A}$-valued inner product.

The reproducing kernel Hilbert $\mathcal{A}$-module (RKHM) associated with $k$
is defined as the completion of $\mathcal{M}_{k,0}$. We denote by
$\mathcal{M}_{k}$ the RKHM associated with $k$.

Heo (2008) focused on the case where a group acts on $\mathcal{X}$ and
investigated corresponding actions on RKHMs. Moreover, he considered the space
of operators on Hilbert $\mathcal{A}$-module and proved that for each
operator-valued positive definite kernel associated with a group and cocycle,
there is a corresponding representation on the Hilbert $C^{*}$-module
associated with the positive definite kernel.

##  3 Application of RKHM to functional data

In this section, we provide an overview of the motivation for studying RKHM
for data analysis. We especially focus on the application of RKHM to
functional data.

Analyzing functional data has been researched to take advantage of the
additional information implied by the smoothness of functions underlying data
(Ramsay and Silverman, 2005; Levitin et al., 2007; Wang et al., 2016). By
describing data as functions, we obtain information as functions such as
derivatives. Applying kernel methods to functional data is also proposed
(Kadri et al., 2016). In these frameworks, the functions are assumed to be
vectors in a Hilbert space such as $L^{2}(\Omega)$ for a measure space
$\Omega$, or they are embedded in an RKHS or vvRKHS. Then, analyses are
addressed in these Hilbert spaces.

However, since functional data itself is infinite-dimensional data, Hilbert
spaces are not always sufficient for extracting its continuous behavior. This
is because the inner products in Hilbert spaces are complex-valued,
degenerating or failing to capture the continuous behavior of the functional
data. We compare algorithms in Hilbert spaces and those in Hilbert
$C^{*}$-modules and show advantages of algorithms in Hilbert $C^{*}$-modules
over those in Hilbert spaces, which are summarized in Figure 1. We first
consider algorithms in Hilbert spaces for analyzing functional data
$x_{1},x_{2},\ldots\in C(\Omega,\mathcal{X})$, where $\Omega$ is a
$\sigma$-finite measure space and $\mathcal{X}$ is a Hilbert space. There are
two possible typical patterns of algorithms in Hilbert spaces. The first
pattern (Pattern 1 in Fig. 1) is regarding each function $x_{i}$ as a vector
in a Hilbert space $\mathcal{H}$ containing $C(\Omega,\mathcal{X})$. In this
case, the inner product $\left\langle x_{i},x_{j}\right\rangle_{\mathcal{H}}$
between two functions $x_{i}$ and $x_{j}$ is single complex-valued although
$x_{i}$ and $x_{j}$ are functions. Therefore, information of the value of
functions at each point degenerates into a complex value. The second pattern
(Pattern 2 in Fig. 1) is discretizing each function $x_{i}$ as
$x_{i}(t_{0}),x_{i}(t_{1}),\ldots$ for $t_{0},t_{1},\ldots\in\Omega$ and
regarding each discretized value $x_{i}(t_{l})$ as a vector in the Hilbert
space $\mathcal{X}$. In this case, we obtain the complex-valued inner product
$\left\langle x_{i}(t_{l}),x_{j}(t_{l})\right\rangle_{\mathcal{X}}$ at each
point $t_{l}\in\Omega$. However, because of the discretization, continuous
behaviors, for example, derivatives, total variation, and frequency
components, of the function $x_{i}$ are lost. Algorithms of both patterns in
the Hilbert spaces proceed by using the computed complex-valued inner
products. As a result, capturing features of functions with the algorithms in
the Hilbert spaces is difficult. On the other hand, if we regard each function
$x_{i}$ as a vector in a Hilbert $C^{*}$-module $\mathcal{M}$ (the rightmost
picture in Fig. 1), then the inner product $\left\langle
x_{i},x_{j}\right\rangle_{\mathcal{M}}$ between two functions $x_{i}$ and
$x_{j}$ in the Hilbert $C^{*}$-module is $C^{*}$-algebra-valued. Thus, if we
set the $C^{*}$-algebra as a function space such as $L^{\infty}(\Omega)$, the
inner product $\left\langle x_{i},x_{j}\right\rangle_{\mathcal{M}}$ is
function-valued. Therefore, algorithms in Hilbert $C^{*}$-modules enable us to
capture and extract continuous behaviors of functions. Moreover, in the case
of the outputs are functions, we can control the outputs according to the
features of the functions.

Since RKHM is a generalization of RKHS and vvRKHS (see Subsection 4.2 for
further details), the framework of RKHMs (Hilbert $C^{*}$-modules) allows us
to generalize kernel methods in RKHSs and vvRKHSs (Hilbert spaces) to those in
Hilbert $C^{*}$-modules. Therefore, by using RKHM, we can capture and extract
features of functions in kernel methods. The remainder of this paper is
devoted to developing the theory of applying RKHMs to data analysis and
showing examples of practical applications of data analysis in RKHMs (PCA,
time-series data analysis, and analysis of interaction effects).

$x_{2}(t)$$t$$x_{1}(t)$$t$Compute theinner product$\left\langle
x_{1},x_{2}\right\rangle_{\mathcal{H}}\in\mathbb{C}$$c_{i}=\left\langle
x_{1}(t_{i}),x_{2}(t_{i})\right\rangle\in\mathbb{C}$Degenerates
informationalong $t$aa$x_{1}$,$x_{2}$ : Functional
data$x_{1},x_{2}\in\mathcal{H}$Algorithms in Hilbert spaces(e.g. RKHSs and
vvRKHSs)(Pattern 1)

$x_{2}(t)$$t$$x_{1}(t)$$t$$t_{0}$$t_{1}$$t_{2}$$t_{3}$$t_{4}$$t_{5}$$t_{6}$$t_{7}$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{5}$$c_{6}$$c_{7}$$c_{i}=\left\langle
x_{1}(t_{i}),x_{2}(t_{i})\right\rangle_{\mathcal{X}}\in\mathbb{C}$Fails to
capturecontinuous behaviors(derivatives, total variation,frequency
components,…)Functional dataAlgorithms(e.g. RKHSs)(Pattern
2)$x_{1}(t),x_{2}(t)\in\mathcal{X}$

$x_{2}(t)$$t$$x_{1}(t)$$t$$\left\langle
x_{1},x_{2}\right\rangle(t)$$t$$\left\langle
x_{1},x_{2}\right\rangle_{\mathcal{M}}\in\mathcal{A}$Capture and
controlcontinuous behaviorsFunctional dataAlgorithms inHilbert
$C^{*}$-modules(e.g. RKHMs)$x_{1},x_{2}\in\mathcal{M}$

Figure 1: Advantages of algorithms in Hilbert $C^{*}$-modules over those in
Hilbert spaces

##  4 RKHM for data analysis

As we mentioned in Section 1, RKHM has been studied in mathematical physics
and pure mathematics. In existing studies, mathematical properties of RKHM
such as the relationship between group actions and RKHMs (see the last
paragraph of Subsection 2.5) have been discussed. However, these studies have
not been focused on data and algorithms for analyzing it. Therefore, we fill
the gaps between the existing theory of RKHM and its application to data
analysis in this section. We develop theories for the validity to applying it
to data analysis in Subsection 4.1. Also, we investigate the connection of
RKHM with RKHS and vvRKHS in Subsection 4.2.

Generalizations of theories of Hilbert space and RKHS are quite nonobvious for
general $C^{*}$-algebras since fundamental properties in Hilbert spaces such
as the Riesz representation theorem and orthogonal complementedness are not
always obtained in Hilbert $C^{*}$-modules. Therefore, we consider limiting
$C^{*}$-algebras to an appropriate class of $C^{*}$-algebras. In fact, von
Neumann-algebras satisfy desired properties.

######  Definition 4.1 (von Neumann-algebra)

A C*-algebra $\mathcal{A}$ is called a von Neumann-algebra if $\mathcal{A}$ is
isomorphic to the dual Banach space of some Banach space.

The following propositions are fundamental for deriving useful properties for
data analysis in Hilbert $C^{*}$-modules and RKHMs (Skeide, 2000, Theorem
4.16), (Manuilov and Troitsky, 2000, Proposition 2.3.3).

######  Proposition 4.2 (The Riesz representation theorem for Hilbert
$\mathcal{A}$-modules)

Let
$\mathcal{A}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\subseteq\mathcal{B}(\mathcal{W})}$
be a von Neumann-algebra and let $\mathcal{M}$ be a Hilbert
$\mathcal{A}$-module. Let
$\mathcal{H}=\mathcal{M}\otimes_{\mathcal{A}}\mathcal{W}$ (see Definition 4.12
for the definition of the product
$\mathcal{M}\otimes_{\mathcal{A}}\mathcal{W}$). Then, every $v\in\mathcal{M}$
can be regarded as an operator in $\mathcal{B}(\mathcal{W},\mathcal{H})$, the
set of bounded linear operators from $\mathcal{W}$ to $\mathcal{H}$. If
$\mathcal{M}\subseteq\mathcal{B}(\mathcal{W},\mathcal{H})$ is strongly closed
(in this case, we say that $\mathcal{M}$ is a von Neumann
$\mathcal{A}$-module), then for a bounded $\mathcal{A}$-linear map
$L:\mathcal{M}\to\mathcal{A}$ (see Definition 2.19), there exists a unique
$u\in\mathcal{M}$ such that $Lv=\left\langle u,v\right\rangle_{\mathcal{M}}$
for all $v\in\mathcal{M}$.

Let $\mathcal{A}$ be a von Neumann-algebra. We remark that the Hilbert
$\mathcal{A}$-module $\mathcal{A}^{n}$ for some $n\in\mathbb{N}$ is a von
Neumann $\mathcal{A}$-module. Moreover, for an $\mathcal{A}$-valued positive
definite kernel defined as $\tilde{k}1_{\mathcal{A}}$, where $\tilde{k}$ is a
(standard) positive definite kernel, the RKHM $\mathcal{M}_{k}$ is a von
Neumann $\mathcal{A}$-module. (Generally, the Hilbert $\mathcal{A}$-module
represented as $\mathcal{H}\otimes\mathcal{A}$ for a Hilbert space
$\mathcal{H}$ is a von Neumann $\mathcal{A}$-module. Here, $\otimes$
represents the tensor product of a Hilbert space and $C^{*}$-module. See Lance
(1995, p.6) for further details about the tensor product.)

######  Proposition 4.3 (Orthogonal complementedness in Hilbert
$\mathcal{A}$-modules)

Let $\mathcal{A}$ be a unital $C^{*}$-algebra and let $\mathcal{M}$ be a
Hilbert $\mathcal{A}$-module. Let $\mathcal{V}$ be a finitely (algebraically)
generated closed submodule of $\mathcal{M}$. Then, any $u\in\mathcal{M}$ is
decomposed into $u=u_{1}+u_{2}$ where $u_{1}\in\mathcal{V}$ and
$u_{2}\in\mathcal{V}^{\perp}$. Here, $\mathcal{V}^{\perp}$ is the orthogonal
complement of $\mathcal{V}$ defined as $\\{u\in\mathcal{M}\mid\ \left\langle
u,v\right\rangle_{\mathcal{M}}=0\\}$.

Let $\mathcal{A}$ be a unital $C^{*}$-algebra, let $\mathcal{M}$ be a Hilbert
$\mathcal{A}$-module, and let $\\{q_{1},\ldots,q_{n}\\}$ be an ONS of
$\mathcal{M}$. Then, the submodule $\mathcal{V}$ generated by
$\\{q_{1},\ldots,q_{n}\\}$ is isomorphic to
$\bigoplus_{i=1}^{n}\mathcal{V}_{i}$, where $\mathcal{V}_{i}=\\{\left\langle
q_{i},q_{i}\right\rangle_{\mathcal{M}}c\mid\ c\in\mathcal{A}\\}$ is a closed
submodule of $\mathcal{A}$. Thus, we have
$\mathcal{M}=\mathcal{V}\oplus\mathcal{V}^{\perp}$.

Therefore, we set $\mathcal{A}$ as a von Neumann-algebra to derive useful
properties of RKHM for data analysis. Note that every von Neumann-algebra is
unital (see Definition 2.5).

###### Assumption 4.4

We assume $\mathcal{A}$ is a von Neumann-algebra throughout this paper.

$C^{*}$-algebras in Example 2.6 are also von Neumann-algebras. As we noted
after Example 2.6, any $C^{*}$-algebra can be regarded as a subalgebra of
$\mathcal{B}(\mathcal{W})$. Thus, this fact implies setting the range of the
positive definite kernel as $\mathcal{B}(\mathcal{W})$ rather than general
$C^{*}$-algebras is effective for data analysis.

###  4.1 General properties of RKHM for data analysis

####  4.1.1 Fundamental properties of RKHM

Similar to the cases of RKHSs, we show RKHMs constructed by
$\mathcal{A}$-valued positive definite kernels have the reproducing property.
Also, we show that the RKHM associated with an $\mathcal{A}$-valued positive
definite kernel $k$ is uniquely determined.

###### Proposition 4.5

The map $\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}_{k}}$ defined on
$\mathcal{M}_{k,0}$ is extended continuously to $\mathcal{M}_{k}$ and the map
$\mathcal{M}_{k}\ni
v\mapsto(x\mapsto\left\langle\phi(x),v\right\rangle_{\mathcal{M}_{k}})\in\mathcal{A}^{\mathcal{X}}$
is injective. Thus, $\mathcal{M}_{k}$ is regarded to be the subset of
$\mathcal{A}^{\mathcal{X}}$ and has the reproducing property.

###### Proposition 4.6

Assume a Hilbert $C^{*}$-module $\mathcal{M}$ over $\mathcal{A}$ and a map
$\psi:\mathcal{X}\to\mathcal{M}$ satisfy the following conditions:

  1. 1.

${}^{\forall}x,y\in\mathcal{X}$,
$\left\langle\psi(x),\psi(y)\right\rangle_{\mathcal{M}}=k(x,y)$

  2. 2.

$\overline{\\{\sum_{i=1}^{n}\psi(x_{i})c_{i}\mid\ x_{i}\in\mathcal{X},\
c_{i}\in\mathcal{A}\\}}=\mathcal{M}$

Then, there exists a unique $\mathcal{A}$-linear bijection map
$\Psi:\mathcal{M}_{k}\to\mathcal{M}$ that preserves the inner product and
satisfies the following commutative diagram:

$\textstyle{\mathcal{M}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\mathcal{M}}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\scriptstyle{\circlearrowright}$

We give the proofs for the above propositions in Appendix A.

####  4.1.2 Minimization property and representer theorem in RKHMs

We now develop some theories for the validity to apply RKHM to data analysis.
First, we show a minimization property of orthogonal projection operators,
which is a fundamental property in Hilbert spaces, is also available in
Hilbert $C^{*}$-modules.

######  Theorem 4.7 (Minimization property of orthogonal projection operators)

Let $\mathcal{I}$ be a finite index set. Let $\\{q_{i}\\}_{i\in\mathcal{I}}$
be an ONS of $\mathcal{M}$ and $\mathcal{V}$ be the submodule of $\mathcal{M}$
spanned by $\\{q_{i}\\}_{i\in\mathcal{I}}$. For $u\in\mathcal{M}_{k}$, let
$P:\mathcal{M}\to\mathcal{V}$ be the projection operator defined as
$Pu:=\sum_{i\in\mathcal{I}}q_{i}\left\langle
q_{i},u\right\rangle_{\mathcal{M}}$. Then $Pu$ is the unique solution of the
following minimization problem, where the minimum is taken with respect to a
(pre) order in $\mathcal{A}$ (see Definition 2.9):

$\min_{v\in\mathcal{V}}|u-v|_{\mathcal{M}}^{2}.$ (2)

Proof  By Proposition 4.3, $u\in\mathcal{M}$ is decomposed into
$u=u_{1}+u_{2}$, where $u_{1}=Pu\in\mathcal{V}$ and
$u_{2}=u-u_{1}\in\mathcal{V}^{\perp}$. Let $v\in\mathcal{V}$. Since
$u_{1}-v\in\mathcal{V}$, the identity $\left\langle
u_{2},u_{1}-v\right\rangle_{\mathcal{M}}=0$ holds. Therefore, we have

$|u-v|_{\mathcal{M}}^{2}=|u_{2}+(u_{1}-v)|_{\mathcal{M}}^{2}=|u_{2}|_{\mathcal{M}}^{2}+|u_{1}-v|_{\mathcal{M}}^{2},$
(3)

which implies
$|u-v|_{\mathcal{M}}^{2}-|u-u_{1}|_{\mathcal{M}}^{2}\geq_{\mathcal{A}}0$.
Since $v\in\mathcal{V}$ is arbitrary, $u_{1}$ is a solution of
$\min_{v\in\mathcal{V}}|u-v|_{\mathcal{M}}$.

Moreover, if there exists $u^{\prime}\in\mathcal{V}$ such that
$|u-u_{1}|_{\mathcal{M}}^{2}=|u-u^{\prime}|_{\mathcal{M}}^{2}$, then letting
$v=u^{\prime}$ in Eq. (3) derives
$|u-u^{\prime}|_{\mathcal{M}}^{2}=|u_{2}|_{\mathcal{M}}^{2}+|u_{1}-u^{\prime}|_{\mathcal{M}}^{2}$,
which implies $|u_{1}-u^{\prime}|_{\mathcal{M}}^{2}=0$. As a result,
$u_{1}=u^{\prime}$ holds and the uniqueness of $u_{1}$ has been proved.
Proposition 4.7 shows the orthogonally projected vector uniquely minimizes the
deviation from an original vector in $\mathcal{V}$. Thus, we can generalize
methods related to orthogonal projections in Hilbert spaces to Hilbert
$C^{*}$-modules.

Next, we show the representer theorem in RKHMs.

######  Theorem 4.8 (Representer theorem)

Let $x_{1},\ldots,x_{n}\in\mathcal{X}$ and $a_{1},\ldots,a_{n}\in\mathcal{A}$.
Let $h:\mathcal{X}\times\mathcal{A}^{2}\to\mathcal{A}_{+}$ be an error
function and let $g:\mathcal{A}_{+}\to\mathcal{A}_{+}$ satisfy
$g(c)\leq_{\mathcal{A}}g(d)$ for $c\leq_{\mathcal{A}}d$. Assume the module
spanned by $\\{\phi(x_{i})\\}_{i=1}^{n}$ is closed. Then, any
$u\in\mathcal{M}_{k}$ minimizing
$\sum_{i=1}^{n}h(x_{i},a_{i},u(x_{i}))+g(|u|_{\mathcal{M}_{k}})$ admits a
representation of the form $\sum_{i=1}^{n}\phi(x_{i})c_{i}$ for some
$c_{1},\ldots,c_{n}\in\mathcal{A}$.

Proof  Let $\mathcal{V}$ be the module spanned by
$\\{\phi(x_{i})\\}_{i=1}^{n}$. By Proposition 4.3, $u\in\mathcal{M}_{k}$ is
decomposed into $u=u_{1}+u_{2}$, where $u_{1}\in\mathcal{V}$,
$u_{2}\in\mathcal{V}^{\perp}$. By the reproducing property of
$\mathcal{M}_{k}$, the following equalities are derived for $i=1,\ldots,n$:

$\displaystyle
u(x_{i})=\left\langle\phi(x_{i}),u\right\rangle_{\mathcal{M}_{k}}=\left\langle\phi(x_{i}),u_{1}+u_{2}\right\rangle_{\mathcal{M}_{k}}=\left\langle\phi(x_{i}),u_{1}\right\rangle_{\mathcal{M}_{k}}.$

Thus, $\sum_{i=1}^{n}h(x_{i},a_{i},u(x_{i}))$ is independent of $u_{2}$. As
for the term $g(|u|_{\mathcal{M}_{k}})$, since $g$ satisfies
$g(c)\leq_{\mathcal{A}}g(d)$ for $c\leq_{\mathcal{A}}d$, we have

$\displaystyle
g(|u|_{\mathcal{M}_{k}})=g(|u_{1}+u_{2}|_{\mathcal{M}_{k}})=g\Big{(}\big{(}|u_{1}|_{\mathcal{M}_{k}}^{2}+|u_{2}|_{\mathcal{M}_{k}}^{2}\big{)}^{1/2}\Big{)}\geq_{\mathcal{A}}g(|u_{1}|_{\mathcal{M}_{k}}).$

Therefore, setting $u_{2}=0$ does not affect the term
$\sum_{i=1}^{n}h(x_{i},a_{i},u(x_{i}))$, while strictly reducing the term
$g(|u|_{\mathcal{M}_{k}})$, which implies any minimizer must have $u_{2}=0$.
As a result, any minimizer takes the form $\sum_{i=1}^{n}\phi(x_{i})c_{i}$.

###  4.2 Connection with RKHSs and vvRKHSs

We show that the framework of RKHM is more general than those of RKHS and
vvRKHS. Let $\tilde{k}$ be a complex-valued positive definite kernel and let
$\mathcal{H}_{\tilde{k}}$ be the RKHS associated with $\tilde{k}$. In
addition, let $k$ be an $\mathcal{A}$-valued positive definite kernel and
$\mathcal{M}_{k}$ be the RKHM associated with $k$. The following proposition
is derived by the definitions of RKHSs and RKHMs.

######  Proposition 4.9 (Connection between RKHMs with RKHSs)

If $\mathcal{A}=\mathbb{C}$ and $k=\tilde{k}$, then
$\mathcal{H}_{\tilde{k}}=\mathcal{M}_{k}$.

As for the connection between vvRKHSs and RKHMs, we first remark that in the
case of $\mathcal{A}=\mathcal{B}(\mathcal{W})$, Definition 2.21 is equivalent
to the operator valued positive definite kernel (Definition 2.2) for the
theory of vv-RKHSs.

######  Lemma 4.10 (Connection between Definition 2.21 and Definition 2.2)

If $\mathcal{A}=\mathcal{B}(\mathcal{W})$, then, the $\mathcal{A}$-valued
positive definite kernel defined in Definition 2.21 is equivalent to the
operator valued positive definite kernel defined in Definition 2.2.

The proof for Lemma 4.10 is given in Appendix A.

Let $\mathcal{A}=\mathcal{B}(\mathcal{W})$ and let
$\mathcal{H}_{k}^{\operatorname{v}}$ be the vvRKHS associated with $k$. To
investigate further connections between vvRKHSs and RKHMs, we introduce the
notion of interior tensor (Lance, 1995, Chapter 4).

###### Proposition 4.11

Let $\mathcal{M}$ be a Hilbert $\mathcal{B}(\mathcal{W})$-module and let
$\mathcal{M}\otimes\mathcal{W}$ be the tensor product of $\mathcal{M}$ and
$\mathcal{W}$ as vector spaces. The map
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}\otimes\mathcal{W}}:\mathcal{M}\otimes\mathcal{W}\;\times\;\mathcal{M}\otimes\mathcal{W}\to\mathbb{C}$
defined as

$\left\langle v\otimes w,u\otimes
h\right\rangle_{\mathcal{M}\otimes\mathcal{W}}=\left\langle w,\left\langle
v,u\right\rangle_{\mathcal{M}}h\right\rangle_{\mathcal{W}}$

is a complex-valued pre inner product on $\mathcal{M}\otimes\mathcal{W}$.

######  Definition 4.12 (Interior tensor)

The completion of $\mathcal{M}\otimes\mathcal{W}$ with respect to the pre
inner product
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{M}\otimes\mathcal{W}}$ is
referred to as the interior tensor between $\mathcal{M}$ and $\mathcal{W}$,
and denoted as $\mathcal{M}\otimes_{\mathcal{B}(\mathcal{W})}\mathcal{W}$.

Note that $\mathcal{M}\otimes_{\mathcal{B}(\mathcal{W})}\mathcal{W}$ is a
Hilbert space. We now show vvRKHSs are reconstructed by the interior tensor
between RKHMs and $\mathcal{W}$.

######  Theorem 4.13 (Connection between RKHMs and vvRKHSs)

If $\mathcal{A}=\mathcal{B}(\mathcal{W})$, then two Hilbert spaces
$\mathcal{H}_{k}^{\operatorname{v}}$ and
$\mathcal{M}\otimes_{\mathcal{B}(\mathcal{W})}\mathcal{W}$ are isomorphic.

Theorem 4.13 is derived by the following lemma.

###### Lemma 4.14

There exists a unique unitary map
$U\colon\mathcal{M}_{k}\otimes_{\mathcal{B}(\mathcal{W})}\mathcal{W}\to\mathcal{H}_{k}^{\operatorname{v}}$
such that $U(\phi(x)c\otimes w)=\phi(x)(cw)$ holds for all $x\in\mathcal{X}$,
$c\in\mathcal{B}(\mathcal{W})$ and $w\in\mathcal{W}$.

Proof  First, we show that

$\displaystyle\bigg{\langle}\sum_{i=1}^{n}\phi({x_{i}})c_{i}\otimes
w_{i},\sum_{j=1}^{l}\phi({y_{j}})d_{j}\otimes
h_{j}\bigg{\rangle}_{\mathcal{M}_{k}\otimes\mathcal{W}}=\bigg{\langle}\sum_{i=1}^{n}\phi({x_{i}})(c_{i}w_{i}),\sum_{j=1}^{l}\phi({y_{j}})(d_{j}h_{j})\bigg{\rangle}_{\mathcal{H}_{k}^{\operatorname{v}}}$

holds for all $\sum_{i=1}^{n}\phi({x_{i}})c_{i}\otimes
w_{i},\sum_{j=1}^{l}\phi({y_{j}})d_{j}\otimes
h_{j}\in\mathcal{M}_{k}\otimes_{\mathcal{B}(\mathcal{W})}\mathcal{W}$. This
follows from the straightforward calculation. Indeed, we have

$\displaystyle\bigg{\langle}\sum_{i=1}^{n}\phi({x_{i}})c_{i}\otimes
w_{i},\sum_{j=1}^{l}\phi({y_{j}})d_{j}\otimes
h_{j}\bigg{\rangle}_{\mathcal{M}_{k}\otimes\mathcal{W}}=\sum_{i=1}^{n}\sum_{j=1}^{l}\left\langle
w_{i},\left\langle\phi(x_{i})c_{i},\phi(y_{j})d_{j}\right\rangle_{k}h_{j}\right\rangle_{\mathcal{W}}$
$\displaystyle\qquad=\sum_{i=1}^{n}\sum_{j=1}^{l}\left\langle
w_{i},c_{i}^{*}k(x_{i},y_{j})d_{j}h_{j}\right\rangle_{\mathcal{W}}=\sum_{i=1}^{n}\sum_{j=1}^{l}\left\langle
c_{i}w_{i},k(x_{i},y_{j})d_{j}h_{j}\right\rangle_{\mathcal{W}}$
$\displaystyle\qquad=\bigg{\langle}\sum_{i=1}^{n}\phi({x_{i}})(c_{i}w_{i}),\sum_{j=1}^{l}\phi({y_{j}})(d_{j}h_{j})\bigg{\rangle}_{\mathcal{H}_{k}^{\operatorname{v}}}.$

Therefore, by the standard functional analysis argument, it turns out that
there exists an isometry
$U\colon\mathcal{M}_{k}\otimes_{\mathcal{B}(\mathcal{W})}\mathcal{W}\to\mathcal{H}_{k}^{\operatorname{v}}$
such that $U(\phi(x)c\otimes w)=\phi(x)(cw)$ holds for all $x\in\mathcal{X}$,
$c\in\mathcal{B}(\mathcal{W})$ and $w\in\mathcal{W}$. Since the image of $U$
is closed and dense in $\mathcal{H}_{k}^{\operatorname{v}}$, $U$ is
surjective. Thus $U$ is a unitary map.

##  5 Kernel mean embedding in RKHM

We generalize KME in RKHSs, which is widely used in analyzing distributions,
to RKHMs. By using the framework of RKHM, we can embed $\mathcal{A}$-valued
measures instead of probability measures (more generally, complex-valued
measures). We provide a brief review of $\mathcal{A}$-valued measures and the
integral with respect to $\mathcal{A}$-valued measures in Appendix B. We
define a KME in RKHMs in Subsection 5.1 and show its theoretical properties in
Subsection 5.2.

To define a KME by using $\mathcal{A}$-valued measures and integrals, we first
define $c_{0}$-kernels.

######  Definition 5.1 (Function space ${C}_{0}(\mathcal{X},\mathcal{A})$)

For a locally compact Hausdorff space $\mathcal{X}$, the set of all
$\mathcal{A}$-valued continuous functions on $\mathcal{X}$ vanishing at
infinity is denoted as $C_{0}(\mathcal{X},\mathcal{A})$. Here, an
$\mathcal{A}$-valued continuous function $u$ is said to vanish at infinity if
the set $\\{x\in\mathcal{X}\mid\ \|u(x)\|_{\mathcal{A}}\geq\epsilon\\}$ is
compact for any $\epsilon>0$. The space ${C}_{0}(\mathcal{X},\mathcal{A})$ is
a Banach $\mathcal{A}$-module with respect to the sup norm.

Note that if $\mathcal{X}$ is compact, any continuous function is contained in
${C}_{0}(\mathcal{X},\mathcal{A})$.

######  Definition 5.2 ($c_{0}$-kernel)

Let $\mathcal{X}$ be a locally compact Hausdorff space. An
$\mathcal{A}$-valued positive definite kernel
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ is referred to as a
$c_{0}$-kernel if $k$ is bounded and
$\phi(x)=k(\cdot,x)\in{C}_{0}(\mathcal{X},\mathcal{A})$ for any
$x\in\mathcal{X}$.

In this section, we impose the following assumption.

###### Assumption 5.3

We assume $\mathcal{X}$ is a locally compact Hausdorff space and $k$ is an
$\mathcal{A}$-valued $c_{0}$-positive definite kernel. In addition, we assume
$\mathcal{M}_{k}$ is a von Neumann $\mathcal{A}$-module (see Proposition 4.2).

For example, we often consider $\mathcal{X}=\mathbb{R}^{d}$ in practical
situations. Also, we provide examples of $c_{0}$-kernels as follows.

###### Example 5.4

  1. 1.

Let $\mathcal{A}=L^{\infty}([0,1])$ and $k$ is an $\mathcal{A}$-valued
positive definite kernel defined such that $k(x,y)(t)$ is a complex-valued
positive definite kernel for $t\in[0,1]$ (see Example 2.22.2). Assume there
exists a complex-valued $c_{0}$-positive definite kernel $\tilde{k}$ such that
for any $t\in[0,1]$, $|k(x,y)(t)|\leq|\tilde{k}(x,y)|$ holds. If
$\|k(x,y)\|_{\mathcal{A}}$ is continuous with respect to $y$ for any
$x\in\mathcal{X}$, then the inclusion

$\\{y\in\mathcal{X}\mid\
\|k(x,y)\|_{\mathcal{A}}\geq\epsilon\\}\subseteq\\{y\in\mathcal{X}\mid\
|\tilde{k}(x,y)|\geq\epsilon\\}$

holds for $x\in\mathcal{X}$ and $\epsilon>0$. Since $\tilde{k}$ is a
$c_{0}$-kernel, the set $\\{y\in\mathcal{X}\mid\
|\tilde{k}(x,y)|\geq\epsilon\\}$ is compact (see Definition 5.1). Thus,
$\\{y\in\mathcal{X}\mid\ \|k(x,y)\|_{\mathcal{A}}\geq\epsilon\\}$ is also
compact and $k$ is an $\mathcal{A}$-valued $c_{0}$-positive definite kernel.
Examples of complex-valued $c_{0}$-positive definite kernels are Gaussian,
Laplacian and $B_{2n+1}$-spline kernels.

  2. 2.

Let $\mathcal{W}$ be a separable Hilbert space and let
$\\{e_{i}\\}_{i=1}^{\infty}$ be an orthonormal basis of $\mathcal{W}$. Let
$\mathcal{A}=\mathcal{B}(\mathcal{W})$ and let
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be defined as
$k(x,y)e_{i}=k_{i}(x,y)e_{i}$, where
$k_{i}:\mathcal{X}\times\mathcal{X}\to\mathbb{C}$ is a complex-valued positive
definite kernel for any $i=1,2,\ldots$ (see Example 2.22.3). Assume there
exists a complex-valued $c_{0}$-positive definite kernel $\tilde{k}$ such that
for any $i=1,2,\ldots$, $|k_{i}(x,y)|\leq|\tilde{k}(x,y)|$ holds.

If $\|k(x,y)\|_{\mathcal{A}}$ is continuous with respect to $y$ for any
$x\in\mathcal{X}$, then $k$ is shown to be an $\mathcal{A}$-valued
$c_{0}$-positive definite kernel in the same manner as the above example.

We introduce $\mathcal{A}$-valued measure and integral in preparation for
defining a KME in RKHMs. They are special cases of vector measure and integral
(Dinculeanu, 1967, 2000), respectively. We review vector measure and integral
as $\mathcal{A}$-valued ones in Appendix B. The notions of measure and the
Lebesgue integral are generalized to $\mathcal{A}$-valued.

###  5.1 Kernel mean embedding of $C^{*}$-algebra-valued measures

We now define a KME in RKHMs.

######  Definition 5.5 (KME in RKHMs)

Let $\mathcal{D}(\mathcal{X},\mathcal{A})$ be the set of all
$\mathcal{A}$-valued finite regular Borel measures. A kernel mean embedding in
an RKHM $\mathcal{M}_{k}$ is a map
$\Phi:\mathcal{D}(\mathcal{X},\mathcal{A})\rightarrow\mathcal{M}_{k}$ defined
by

$\Phi(\mu):=\int_{x\in\mathcal{X}}\phi(x)d\mu(x).$ (4)

We emphasize that the well-definedness of $\Phi$ is not trivial, and von
Neumann $\mathcal{A}$-module is adequate to show it. More precisely, the
following theorem derives the well-definedness.

######  Theorem 5.6 (Well-definedness for the KME in RKHMs)

Let $\mu\in\mathcal{D}(\mathcal{X},\mathcal{A})$. Then,
$\Phi(\mu)\in\mathcal{M}_{k}$. In addition, the following equality holds for
any $v\in\mathcal{M}_{k}$:

$\left\langle\Phi(\mu),v\right\rangle_{\mathcal{M}_{k}}=\int_{x\in\mathcal{X}}d\mu^{*}(x)v(x).$
(5)

To show Theorem 5.6, we use the Riesz representation theorem for Hilbert
$\mathcal{A}$-modules (Proposition 4.2).

Proof  Let $L_{\mu}:\mathcal{M}_{k}\to\mathcal{A}$ be an $\mathcal{A}$-linear
map defined as $L_{\mu}v:=\int_{x\in\mathcal{X}}d\mu^{*}(x)v(x)$. The
following inequalities are derived by the reproducing property and the
Cauchy–Schwarz inequality (Lemma 2.16):

$\displaystyle\|L_{\mu}v\|_{\mathcal{A}}$
$\displaystyle\leq\int_{x\in\mathcal{X}}\|v(x)\|_{\mathcal{A}}d|\mu|(x)=\int_{x\in\mathcal{X}}\|\left\langle\phi(x),v\right\rangle_{\mathcal{M}_{k}}\|_{\mathcal{A}}d|\mu|(x)$
$\displaystyle\leq\|v\|_{\mathcal{M}_{k}}\int_{x\in\mathcal{X}}\|\phi(x)\|_{\mathcal{M}_{k}}d|\mu|(x)\leq|\mu|(\mathcal{X})\|v\|_{\mathcal{M}_{k}}\sup_{x\in\mathcal{X}}\|\phi(x)\|_{\mathcal{M}_{k}},$
(6)

where the first inequality is easily checked for a step function
$s(x):=\sum_{i=1}^{n}c_{i}\chi_{E_{i}}(x)$ as follows:

$\displaystyle\bigg{\|}\int_{x\in\mathcal{X}}d\mu^{*}(x)s(x)\bigg{\|}_{\mathcal{A}}$
$\displaystyle=\bigg{\|}\sum_{i=1}^{n}\mu(E_{i})^{*}c_{i}\bigg{\|}_{\mathcal{A}}\leq\sum_{i=1}^{n}\|\mu(E_{i})\|_{\mathcal{A}}\|c_{i}\|_{\mathcal{A}}$
$\displaystyle\leq\sum_{i=1}^{n}|\mu|(E_{i})\|c_{i}\|_{\mathcal{A}}=\int_{x\in\mathcal{X}}\|s(x)\|_{\mathcal{A}}d|\mu|(x).$

Thus, it holds for any totally measurable functions. Since both
$|{\mu}|(\mathcal{X})$ and
$\sup_{x\in\mathcal{X}}\|\phi(x)\|_{\mathcal{M}_{k}}$ are finite, inequality
(6) means $L_{\mu}$ is bounded. Thus, by the Riesz representation theorem for
Hilbert $\mathcal{A}$-modules (Proposition 4.2), there exists
$u_{\mu}\in\mathcal{M}_{k}$ such that $L_{\mu}v=\left\langle
u_{\mu},v\right\rangle_{\mathcal{M}_{k}}$. By setting $v=\phi(y)$, for
$y\in\mathcal{X}$, we have
$u_{\mu}(y)=L_{\mu}\phi(y)^{*}=\int_{x\in\mathcal{X}}k(y,x)d\mu(x)$.
Therefore, $\Phi(\mu)=u_{\mu}\in\mathcal{M}_{k}$ and
$\left\langle\Phi(\mu),v\right\rangle_{\mathcal{M}_{k}}=\int_{x\in\mathcal{X}}d\mu^{*}(x)v(x)$.

###### Corollary 5.7

For $\mu,\nu\in\mathcal{D}(\mathcal{X},\mathcal{A})$, the inner product
between $\Phi(\mu)$ and $\Phi(\nu)$ is given as follows:

$\left\langle\Phi(\mu),\Phi(\nu)\right\rangle_{\mathcal{M}_{k}}=\int_{x\in\mathcal{X}}\int_{y\in\mathcal{X}}d\mu^{*}(x)k(x,y)d\nu(y).$

Moreover, many basic properties for the existing KME in RKHS are generalized
to the proposed KME as follows.

######  Proposition 5.8 (Basic properties of the KME $\Phi$)

For $\mu,\nu\in\mathcal{D}(\mathcal{X},\mathcal{A})$ and $c\in\mathcal{A}$,
$\Phi(\mu+\nu)=\Phi(\mu)+\Phi(\nu)$ and $\Phi(\mu c)=\Phi(\mu)c$ (i.e., $\Phi$
is $\mathcal{A}$-linear, see Definition 2.19) hold. In addition, for
$x\in\mathcal{X}$, $\Phi(\delta_{x})=\phi(x)$ (see Definition B.2 for the
definition of the $\mathcal{A}$-valued Dirac measure $\delta_{x}$).

This is derived from Eqs. (4) and (5). Note that if $\mathcal{A}=\mathbb{C}$,
then the proposed KME (4) is equivalent to the existing KME in RKHS considered
in Sriperumbudur et al. (2011).

###  5.2 Injectivity and universality

Here, we show the connection between the injectivity of the KME and the
universality of RKHM. The proofs of the propositions in this subsection are
given in Appendix C.

####  5.2.1 injectivity

In practice, the injectivity of $\Phi$ is important to transform problems in
$\mathcal{D}(\mathcal{X},\mathcal{A})$ into those in $\mathcal{M}_{k}$. This
is because if a KME $\Phi$ in an RKHM is injective, then $\mathcal{A}$-valued
measures are embedded into $\mathcal{M}_{k}$ through $\Phi$ without loss of
information. Note that, for probability measures, the injectivity of the
existing KME is also referred to as the “characteristic” property. The
injectivity of the existing KME in RKHS has been discussed in, for example,
Fukumizu et al. (2007); Sriperumbudur et al. (2010, 2011). These studies give
criteria for the injectivity of the KMEs associated with important complex-
valued kernels such as transition invariant kernels and radial kernels.
Typical examples of these kernels are Gaussian, Laplacian, and inverse
multiquadratic kernels. Here, we define the transition invariant kernels and
radial kernels for $\mathcal{A}$-valued measures, and generalize their
criteria to RKHMs associated with $\mathcal{A}$-valued kernels.

To characterize transition invariant kernels, we first define a Fourier
transform and support of an $\mathcal{A}$-valued measure.

######  Definition 5.9 (Fourier transform and support of an
$\mathcal{A}$-valued measure)

For an $\mathcal{A}$-valued measure $\lambda$ on $\mathbb{R}^{d}$, the Fourier
transform of $\lambda$, denoted as $\hat{\lambda}$, is defined as

$\hat{\lambda}(x)=\int_{\omega\in\mathbb{R}^{d}}e^{-\sqrt{-1}x^{T}\omega}d\lambda(\omega).$

In addition, the support of $\lambda$ is defined as

$\operatorname{supp}(\lambda)=\\{x\in\mathbb{R}^{d}\mid\
\lambda(\mathcal{U})>_{\mathcal{A}}0\mbox{ for any open set $\mathcal{U}$ such
that }x\in\mathcal{U}\\}.$

######  Definition 5.10 (Transition invariant kernel and radial kernel)

  1. 1.

An $\mathcal{A}$-valued positive definite kernel
$k:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathcal{A}$ is called a transition
invariant kernel if it is represented as $k(x,y)=\hat{\lambda}(y-x)$ for a
positive $\mathcal{A}$-valued measure $\lambda$.

  2. 2.

An $\mathcal{A}$-valued positive definite kernel
$k:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathcal{A}$ is called a radial kernel
if it is represented as $k(x,y)=\int_{[0,\infty)}e^{-t\|x-y\|^{2}}d\eta(t)$
for a positive $\mathcal{A}$-valued measure $\eta$.

Here, an $\mathcal{A}$-valued measure $\mu$ is said to be positive if
$\mu(E)\geq_{\mathcal{A}}0$ for any Borel set $E$.

We show transition invariant kernels and radial kernels induce injective KMEs.

######  Proposition 5.11 (The injectivity for transition invariant kernels)

Let $\mathcal{A}=\mathbb{C}^{m\times m}$ and $\mathcal{X}=\mathbb{R}^{d}$.
Assume $k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ is a transition
invariant kernel with a positive $\mathcal{A}$-valued measure $\lambda$ that
satisfies $\operatorname{supp}(\lambda)=\mathcal{X}$. Then, KME
$\Phi:\mathcal{D}(\mathcal{X},\mathcal{A})\to\mathcal{M}_{k}$ defined as Eq.
(4) is injective.

######  Proposition 5.12 (The injectivity for radial kernels)

Let $\mathcal{A}=\mathbb{C}^{m\times m}$ and $\mathcal{X}=\mathbb{R}^{d}$.
Assume $k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ is a radial kernel with
a positive definite $\mathcal{A}$-valued measure $\eta$ that satisfies
$\operatorname{supp}(\eta)\neq\\{0\\}$. Then, KME
$\Phi:\mathcal{D}(\mathcal{X},\mathcal{A})\to\mathcal{M}_{k}$ defined as Eq.
(4) is injective.

###### Example 5.13

  1. 1.

If $k:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{C}^{m\times m}$ is a
matrix-valued kernel whose diagonal elements are Gaussian, Laplacian, or
$B_{2n+1}$-spline and nondiagonal elements are $0$, then $k$ is a
$c_{0}$-kernel (See Example 2.22.1). There exists a matrix-valued measure
$\lambda$ that satisfies $k(x,y)=\hat{\lambda}(y-x)$ and whose diagonal
elements are nonnegative and supported by $\mathbb{R}^{d}$ (c.f. Table 2 in
Sriperumbudur et al. (2010)) and nondiagonal elements are $0$. Thus, by
Proposition 5.11, $\Phi$ is injective.

  2. 2.

If $k$ is a matrix-valued kernel whose diagonal elements are inverse
multiquadratic and nondiagonal elements are $0$, then $k$ is a $c_{0}$-kernel.
There exists a matrix-valued measure $\eta$ that satisfies
$k(x,y)=\int_{[0,\infty)}e^{-t\|x-y\|^{2}}d\eta(t)$, and whose diagonal
elements are nonnegative and $\operatorname{supp}(\eta)\neq\\{0\\}$ and
nondiagonal elements are $0$ (c.f. Theorem 7.15 in Wendland (2004)). Thus, by
Proposition 5.12, $\Phi$ is injective.

####  5.2.2 Connection with universality

Another important property for kernel methods is universality, which ensures
that kernel-based algorithms approximate each continuous target function
arbitrarily well. For RKHS, Sriperumbudur et al. (2011) showed the equivalence
of the injectivity of the existing KME in RKHSs and universality of RKHSs. We
define a universality of RKHMs as follows.

######  Definition 5.14 (Universality)

An RKHM is said to be universal if it is dense in
${C}_{0}(\mathcal{X},\mathcal{A})$.

We show the above equivalence holds also for RKHM in the case of
$\mathcal{A}=\mathbb{C}^{m\times m}$.

######  Proposition 5.15 (Equivalence of the injectivity and universality for
$\mathcal{A}=\mathbb{C}^{m\times m}$)

Let $\mathcal{A}=\mathbb{C}^{m\times m}$. Then,
$\Phi:\mathcal{D}(\mathcal{X},\mathcal{A})\to\mathcal{M}_{k}$ is injective if
and only if $\mathcal{M}_{k}$ is dense in ${C}_{0}(\mathcal{X},\mathcal{A})$.

By Proposition 5.15, if $k$ satisfies the condition in Proposition 5.11 or
5.12, then $\mathcal{M}_{k}$ is universal.

For the case where $\mathcal{A}$ is infinite dimensional, the universality of
$\mathcal{M}_{k}$ in ${C}_{0}(\mathcal{X},\mathcal{A})$ is a sufficient
condition for the injectivity of the proposed KME.

######  Theorem 5.16 (Connection between the injectivity and universality for
general $\mathcal{A}$)

If $\mathcal{M}_{k}$ is dense in ${C}_{0}(\mathcal{X},\mathcal{A})$, then
$\Phi:\mathcal{D}(\mathcal{X},\mathcal{A})\to\mathcal{M}_{k}$ is injective.

However, the equivalence of the injectivity and universality, and the
injectivity for transition invariant kernels and radial kernels are open
problems. This is because their proofs strongly depend on the Hahn–Banach
theorem and Riesz–Markov representation theorem, and generalizations of these
theorems to $\mathcal{A}$-valued functions and measures are challenging
problems due to the situation peculiar to the infinite dimensional spaces.
Further details of the proofs of propositions in this section are given in
Appendix C.

##  6 Applications

We apply the framework of RKHM described in Sections 4 and 5 to problems in
data analysis. We propose kernel PCA in RKHMs in Subsection 6.1, time-series
data analysis in RKHMs in Subsection 6.2, and analysis of interaction effects
in finite or infinite dimensional data with the proposed KME in RKHMs in
Subsection 6.3. Then, we discuss further applications in Subsection 6.4.

###  6.1 PCA in RKHMs

Principal component analysis (PCA) is a fundamental tool for describing data
in a low dimensional space. Its implementation in RKHSs has also been proposed
(c.f. Schölkopf and Smola (2001)). It enables us to deal with the
nonlinearlity of data by virtue of the high expressive power of RKHSs. Here,
we generalize the PCA in RKHSs to capture more information in data, such as
multivariate data and functional data, by using the framework of RKHM.

##### Applying RKHM to PCA

In the existing framework of PCA in Hilbert spaces, the following
reconstruction error is minimized with respect to vectors
$p_{1},\ldots,p_{r}$:

$\sum_{i=1}^{n}\bigg{\|}x_{i}-\sum_{j=1}^{r}p_{j}\left\langle
p_{j},x_{i}\right\rangle\bigg{\|}^{2},$ (7)

where $x_{1},\ldots,x_{n}$ are given samples in a Hilbert space and
$p_{1},\ldots,p_{r}$ are called principal axes. Here, the complex-valued inner
product $\left\langle p_{j},x_{i}\right\rangle$ is the weight with respect to
the principal axis $p_{j}$ for representing the sample $x_{i}$. PCA for
functional data (functional PCA) has also investigated (Ramsay and Silverman,
2005). For example, in standard functional PCA settings, we set the Hilbert
space as $L^{2}(\Omega)$ for a $\sigma$-finite measure space $\Omega$.
However, if samples $x_{1},\ldots,x_{n}$ are finite dimensional vectors or
functions, Eq. (7) fails to describe their element wise or continuous
dependencies on the principal axes. For $d$-dimensional (finite dimensional)
vectors, we can just split $x_{i}=[x_{i,1},\ldots,x_{i,d}]$ into $d$ vectors
$[x_{i,1},0,\ldots,0],\ldots,[0,\ldots,0,x_{i,d}]$. Then, we can understand
which element is dominant for representing $x_{i}$ by using the principal axis
$p_{j}$. On the other hand, for functional data, the situation is completely
different. For example, assume samples are in $L^{2}(\Omega)$. Since delta
functions are not contained in $L^{2}(\Omega)$, we cannot split a sample
$x_{i}=x_{i}(t)$ into discrete functions. In this case, how can we understand
the continuous dependencies on the principal axes with respect to the variable
$t\in\Omega$? One possible way to answer this question is to employ Hilbert
$C^{*}$-modules instead of Hilbert spaces. We consider the same type of
reconstruction error as Eq. (7) in Hilbert $C^{*}$-modules. In this case, the
inner product $\left\langle p_{j},x_{i}\right\rangle_{\mathcal{W}}$ is
$C^{*}$-algebra-valued, which allows us to provide more information than the
complex-valued one. If we set the $C^{*}$-algebra as the function space on
$\Omega$ such as $L^{\infty}(\Omega)$ and define a $C^{*}$-algebra-valued
inner product which depends on $t\in\Omega$, then, the weight $\left\langle
p_{j},x_{i}\right\rangle_{\mathcal{W}}$ depends on $t$. As a result, we can
extract continuous dependencies of samples on the principal axes. More
generally, PCA is often considered in an RKHS $\mathcal{H}_{\tilde{k}}$. In
this case, $x_{i}$ in Eq. (7) is replaced with $\tilde{\phi}(x_{i})$, where
$\tilde{\phi}$ is the feature map, and the inner product and norm are replaced
with those in the RKHS. We can extract continuous dependencies of samples on
the principal axes by generalizing RKHS to RKHM.

####  6.1.1 Generalization of the PCA in RKHSs to RKHMs

Let $x_{1},\ldots,x_{n}\in\mathcal{X}$ be given samples. Let
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be an $\mathcal{A}$-valued
positive definite kernel on $\mathcal{X}$ and let $\mathcal{M}_{k}$ be the
RKHM associated with $k$. We explore a useful set of axes $p_{1},\ldots,p_{r}$
in $\mathcal{M}_{k}$, which are referred to as principal axes, to describe the
feature of given samples $x_{1},\ldots,x_{n}$. The corresponding components
$p_{j}\left\langle p_{j},\phi(x_{i})\right\rangle_{\mathcal{M}_{k}}$ are
referred to as principal components. We emphasize our proposed PCA in RKHM
provides weights of principal components contained in $\mathcal{A}$, not in
complex numbers. This is a remarkable difference between our method and
existing PCAs. When samples have some structures such as among variables or in
functional data, $\mathcal{A}$-valued weights provide us richer information
than complex-valued ones. For example, if $\mathcal{X}$ is the space of
functions of multi-variables and if we set $\mathcal{A}$ as
$L^{\infty}([0,1])$, then we can reduce multi-variable functional data to
functions in $L^{\infty}([0,1])$, functions of single variable (as illustrated
in Section 6.1.4).

To obtain $\mathcal{A}$-valued weights of principal components, we consider
the following minimization problem regarding the following reconstruction
error (see Definition 2.18 for the definition of ONS):

$\inf_{\\{p_{j}\\}_{j=1}^{r}\subseteq\mathcal{M}_{k}\mbox{\footnotesize:
ONS}}\;\sum_{i=1}^{n}\bigg{|}\phi(x_{i})-\sum_{j=1}^{r}p_{j}\left\langle
p_{j},\phi(x_{i})\right\rangle_{\mathcal{M}_{k}}\bigg{|}_{\mathcal{M}_{k}}^{2},$
(8)

where the infimum is taken with respect to a (pre) order in $\mathcal{A}$ (see
Definition 2.9). Since the identity
$|\phi(x_{i})-\sum_{j=1}^{r}p_{j}\left\langle
p_{j},\phi(x_{i})\right\rangle_{\mathcal{M}_{k}}|_{\mathcal{M}_{k}}^{2}=k(x_{i},x_{i})-\sum_{j=1}^{r}\left\langle\phi(x_{i}),p_{j}\right\rangle_{\mathcal{M}_{k}}\left\langle
p_{j},\phi(x_{i})\right\rangle_{\mathcal{M}_{k}}$ holds and
$\left\langle\phi(x_{i}),p_{j}\right\rangle_{\mathcal{M}_{k}}$ is represented
as $p_{j}(x_{i})$ by the reproducing property, the problem (8) can be reduced
to the minimization problem

$\inf_{\\{p_{j}\\}_{j=1}^{r}\subseteq\mathcal{M}_{k}\mbox{\footnotesize:
ONS}}\;\sum_{i=1}^{n}\sum_{j=1}^{r}-p_{j}(x_{i})p_{j}(x_{i})^{*}.$ (9)

In the case of RKHS, i.e., $\mathcal{A}=\mathbb{C}$, the solution of the
problem (9) is obtained by computing eigenvalues and eigenvectors of Gram
matrices (see, for example, Schölkopf and Smola (2001)). Unfortunately, we
cannot extend their procedure to RKHM straightforwardly. Therefore, we develop
two methods to obtain approximate solutions of the problem (9): by gradient
descents on Hilbert $C^{*}$-modules, and by the minimization of the trace of
the $\mathcal{A}$-valued objective function.

####  6.1.2 Gradient descent on Hilbert $C^{*}$-modules

We propose a gradient descent method on Hilbert $\mathcal{A}$-module for the
case where $\mathcal{A}$ is commutative. An important example of commutative
von Neumann-algebra is $L^{\infty}([0,1])$. The gradient descent for a real-
valued function on a Hilbert space has been proposed (Smyrlis and Zisis,
2004). However, in our situation, the objective function of the problem (9) is
an $\mathcal{A}$-valued function in a Hilbert $C^{*}$-module
$\mathcal{A}^{n}$. Thus, the existing gradient descent is not applicable to
our situation. Therefore, we generalize the existing gradient descent
algorithm to $\mathcal{A}$-valued functions on Hilbert $C^{*}$-modules.

Let $\mathcal{A}$ be a commutative von Neumann-algebra. Assume the positive
definite kernel $k$ takes its values in
$\mathcal{A}_{r}:=\\{c-d\in\mathcal{A}\mid\ c,d\in\mathcal{A}_{+}\\}$. For
example, for $\mathcal{A}=L^{\infty}([0,1])$, $\mathcal{A}_{r}$ is the space
of real-valued $L^{\infty}$ functions on $[0,1]$. Based on the representer
theorem (Theorem 4.8), we find a solution of the problem (9) which is
represented as $p_{j}=\sum_{i=1}^{n}\phi(x_{i})c_{j,i}$ for some
$c_{j,i}\in\mathcal{A}$. Moreover, since $\mathcal{A}$ is commutative,
$p_{j}(x_{i})p_{j}(x_{i})^{*}$ is equal to $p_{j}(x_{i})^{*}p_{j}(x_{i})$.
Therefore, the problem (8) on $\mathcal{M}_{k}$ is equivalent to the following
problem on the Hilbert $\mathcal{A}$-module $\mathcal{A}^{n}$ (see Example
2.15 about $\mathcal{A}^{n}$):

$\inf_{\mathbf{c}_{j}\in\mathcal{A}^{n},\
\\{\sqrt{\mathbf{G}}\mathbf{c}_{j}\\}_{j=1}^{r}\mbox{\footnotesize:
ONS}}-\sum_{j=1}^{r}\mathbf{c}_{j}^{*}\mathbf{G}^{2}\mathbf{c}_{j},$ (10)

where $\mathbf{G}$ is the $\mathcal{A}$-valued Gram matrix defined as
$\mathbf{G}_{i,j}=k(x_{i},x_{j})$. For simplicity, we assume $r=1$, i.e., the
number of principal axes is $1$. We rearrange the problem (10) to the
following problem by adding a penalty term:

$\inf_{\mathbf{c}\in\mathcal{A}^{n}}(-\mathbf{c}^{*}\mathbf{G}^{2}\mathbf{c}+\lambda|\mathbf{c}^{*}\mathbf{Gc}-1_{\mathcal{A}}|_{\mathcal{A}}^{2}),$
(11)

where $\lambda$ is a real positive weight for the penalty term. For $r>1$, let
$\mathbf{c}_{1}$ be a solution of the problem (10). Then, we solve the same
problem in the orthogonal complement of the module spanned by
$\\{\mathbf{c}_{1}\\}$ and set the solution of this problem as
$\mathbf{c}_{2}$. Then, we solve the same problem in the orthogonal complement
of the module spanned by $\\{\mathbf{c}_{1},\mathbf{c}_{2}\\}$ and repeat this
procedure to obtain solutions $\mathbf{c}_{1},\ldots\mathbf{c}_{r}$. The
problem (11) is the minimization problem of an $\mathcal{A}$-valued function
defined on the Hilbert $\mathcal{A}$-module $\mathcal{A}^{n}$. We search a
solution of the problem (11) along the steepest descent directions. To
calculate the steepest descent directions, we introduce a derivative
$Df_{\mathbf{c}}$ of an $\mathcal{A}$-valued function $f$ on a Hilbert
$C^{*}$-module at $\mathbf{c}\in\mathcal{M}$. It is defined as the derivative
on Banach spaces (c.f. Blanchard and Brüning (2015)). The definition of the
derivative is included in Appendix D. The following gives the derivative of
the objective function in problem (11).

######  Proposition 6.1 (Derivative of the objective function)

Let $f:\mathcal{A}^{n}\to\mathcal{A}$ be defined as

$f(\mathbf{c})=-\mathbf{c}^{*}\mathbf{G}^{2}\mathbf{c}+\lambda|\mathbf{c}^{*}\mathbf{G}\mathbf{c}-1_{\mathcal{A}}|_{\mathcal{A}}^{2}.$
(12)

Then, $f$ is infinitely differentiable and the first derivative of $f$ is
calculated as

$Df_{\mathbf{c}}(u)=-2\mathbf{c}^{*}\mathbf{G}^{2}u-4\lambda\mathbf{c}^{*}\mathbf{G}u+4\lambda\mathbf{c}^{*}\mathbf{G}\mathbf{c}\mathbf{c}^{*}\mathbf{G}u.$

Moreover, for each $\mathbf{c}\in\mathcal{A}^{n}$, there exists a unique
$\mathbf{d}\in\mathcal{A}^{n}$ such that
$\left\langle\mathbf{d},u\right\rangle_{\mathcal{A}^{n}}=Df_{\mathbf{c}}(u)$
for any $u\in\mathcal{A}^{n}$. The vector $\mathbf{d}$ is calculated as

$\mathbf{d}=-2\mathbf{G}^{2}\mathbf{c}-4\lambda\mathbf{G}\mathbf{c}+4\lambda\mathbf{G}\mathbf{c}\mathbf{c}^{*}\mathbf{G}\mathbf{c}.$
(13)

Proof  The derivative of $f$ is calculated by the definition and the
assumption that $\mathcal{A}$ is commutative. Since $Df_{\mathbf{c}}$ is a
bounded $\mathcal{A}$-linear operator, by the Riesz representation theorem
(Proposition 4.2), there exists a unique $\mathbf{d}\in\mathcal{A}^{n}$ such
that
$\left\langle\mathbf{d},u\right\rangle_{\mathcal{A}^{n}}=Df_{\mathbf{c}}(u)$.

######  Definition 6.2 (Gradient of $\mathcal{A}$-valued functions on Hilbert
$C^{*}$-modules)

Let $f:\mathcal{M}\to\mathcal{A}$ be a differentiable function. Assume for
each $\mathbf{c}\in\mathcal{M}$, there exists a unique
$\mathbf{d}\in\mathcal{M}$ such that
$\left\langle\mathbf{d},u\right\rangle_{\mathcal{A}^{n}}=Df_{\mathbf{c}}(u)$
for any $u\in\mathcal{M}$. In this case, we denote $\mathbf{d}$ by $\nabla
f_{\mathbf{c}}$ and call it the gradient of $f$ at $\mathbf{c}$.

We now develop an $\mathcal{A}$-valued gradient descent scheme.

###### Theorem 6.3

Assume $f:\mathcal{M}\to\mathcal{A}$ is differentiable. Moreover, assume there
exists $\nabla f_{\mathbf{c}}$ for any $\mathbf{c}\in\mathcal{M}$. Let
$\eta_{t}>0$. Let $\mathbf{c}_{0}\in\mathcal{M}$ and

$\mathbf{c}_{t+1}=\mathbf{c}_{t}-\eta_{t}\nabla f_{\mathbf{c}_{t}}$ (14)

for $t=0,1,\ldots$. Then, we have

$f(\mathbf{c}_{t+1})=f(\mathbf{c}_{t})-\eta_{t}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{M}}+S(\mathbf{c}_{t},\eta_{t}),$ (15)

where $S(\mathbf{c},\eta)$ satisfies $\lim_{\eta\to
0}\|S(\mathbf{c},\eta)\|_{\mathcal{A}}/\eta=0$.

The statement is derived by the definition of the derivative (Definition D.1).
The following examples show the scheme (14) is valid to solve the problem
(11).

###### Example 6.4

Let $\mathcal{A}=L^{\infty}([0,1])$, let $a_{t}=|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}\in\mathcal{A}$ and let
$b_{t,\eta}=S(\mathbf{c}_{t},\eta)\in\mathcal{A}$. If
$a_{t}\geq_{\mathcal{A}}\delta 1_{\mathcal{A}}$ for some positive real value
$\delta$, then the function $a_{t}$ on $[0,1]$ satisfies $a_{t}(s)>0$ for
almost everywhere $s\in[0,1]$. On the other hand, since $b_{t,\eta}$ satisfies
$\lim_{\eta\to 0}\|b_{t,\eta}\|_{\mathcal{A}}/\eta^{2}=0$, there exists
sufficiently small positive real value $\eta_{t,0}$ such that for almost
everywhere $s\in[0,1]$,
$b_{t,\eta_{t,0}}(s)\leq\|b_{t,\eta_{t,0}}\|_{\mathcal{A}}\leq\eta^{2}_{t,0}\delta\leq\eta_{t,0}(1-\xi_{1})\delta$
hold for some positive real value $\xi_{1}$. As a result, $-\eta_{t,0}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}+S(\mathbf{c}_{t},\eta_{t,0})\leq_{\mathcal{A}}-\eta_{t,0}\xi_{1}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}$ holds and by the Eq. (15), we have

$f(\mathbf{c}_{t+1})<_{\mathcal{A}}f(\mathbf{c}_{t})$ (16)

for $t=0,1,\ldots$. As we mentioned in Example 2.8, the inequality (16) means
the function $f(\mathbf{c}_{t+1})\in L^{\infty}([0,1])$ is smaller than the
function $f(\mathbf{c}_{t})\in L^{\infty}([0,1])$ at almost every points on
$[0,1]$, i.e.,

$f(\mathbf{c}_{t+1})(s)<f(\mathbf{c}_{t})(s)$

for almost every $s\in[0,1]$.

###### Example 6.5

Assume $\mathcal{A}$ is a finite dimensional space. If $|\nabla
f_{\mathbf{c}_{t}}|_{\mathcal{A}}^{2}\geq_{\mathcal{A}}\delta 1_{\mathcal{A}}$
for some positive real value $\delta$, the inequality
$f(\mathbf{c}_{t+1})\leq_{\mathcal{A}}f(\mathbf{c}_{t})-\eta_{t}\xi_{1}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}$ holds for $t=0,1,\ldots$ and some
$\eta_{t}$ and $\xi_{1}$ in the same manner as Example 6.4. Moreover, the
function $f$ defined as Eq. (12) is bounded below and $\nabla
f_{\mathbf{c}_{t}}$ is Lipschitz continuous on the set
$\\{\mathbf{c}\in\mathcal{A}^{n}\mid\
f(\mathbf{c})\leq_{\mathcal{A}}f(\mathbf{c}_{0})\\}$. In this case, if there
exists a positive real value $\xi_{2}$ such that $\|\nabla
f_{\mathbf{c}_{t+1}}-\nabla
f_{\mathbf{c}_{t}}\|_{\mathcal{A}^{n}}\geq\xi_{2}\|\nabla
f_{\mathbf{c}_{t}}\|_{\mathcal{A}^{n}}$, then we have

$\xi_{2}\|\nabla f_{\mathbf{c}_{t}}\|_{\mathcal{A}^{n}}\leq
L\|\mathbf{c}_{t+1}-\mathbf{c}_{t}\|_{\mathcal{A}^{n}}\leq L\eta_{t}\|\nabla
f_{\mathbf{c}_{t}}\|_{\mathcal{A}^{n}},$

where $L$ is a Liptschitz constant of $\nabla f_{\mathbf{c}_{t}}$. As a
result, we have

$f(\mathbf{c}_{t+1})\leq_{\mathcal{A}}f(\mathbf{c}_{t})-\eta_{t}\xi_{1}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}\leq_{\mathcal{A}}f(\mathbf{c}_{t})-\frac{\xi_{1}\xi_{2}}{L}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}},$

which implies $\sum_{t=1}^{T}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}\leq_{\mathcal{A}}L/(\xi_{1}\xi_{2})(f(\mathbf{c}_{1})-f(\mathbf{c}_{T+1}))$.
Since $f$ is bounded below, the sum $\sum_{t=1}^{\infty}|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}$ converges. Therefore, $|\nabla
f_{\mathbf{c}_{t}}|^{2}_{\mathcal{A}^{n}}\to 0$ as $t\to\infty$, i.e., the
gradient $\nabla f_{\mathbf{c}_{t}}$ in Eq. (14) converges to $0$.

###### Remark 6.6

It is possible to generalize the above method to the case where the objective
function $f$ has the form $f(\mathbf{c})=\mathbf{c}^{*}\mathbf{G}\mathbf{c}$
for $\mathbf{G}\in\mathcal{A}^{n\times n}$ and $\mathcal{A}$ is
noncommutative. In this case, the derivative $Df_{\mathbf{c}}$ is calculated
as

$Df_{\mathbf{c}}(u)=u^{*}\mathbf{G}\mathbf{c}+\mathbf{c}^{*}\mathbf{G}u.$

Therefore, defining the gradient $\nabla f_{\mathbf{c}}$ as $\nabla
f_{\mathbf{c}}=\mathbf{G}\mathbf{c}$ results in $Df_{\mathbf{c}}(-\eta\nabla
f_{\mathbf{c}})=-2\eta\mathbf{c}^{*}\mathbf{G}^{2}\mathbf{c}\leq_{\mathcal{A}}0$
for a real positive value $\eta$, which allows us to derive the same result as
Theorem 6.3.

###### Remark 6.7

The computational complexity of the PCA in RKHMs is higher than the standard
PCA in RKHSs. Indeed, in the case of RKHSs, the minimization problem is
reduced to an eigenvalue problem of the Gram matrix with respect to given
samples. On the other hand, we solve the minimization problem (8) by the
gradient descent, and in each iteration step, we compute the gradient
$\mathbf{d}$ in Eq. (13). Since the elements of $\mathbf{G}$ and $\mathbf{c}$
are in $\mathcal{A}$, the computation of $\mathbf{d}$ involves the
multiplication in $\mathcal{A}$ such as multiplication of functions. Even
though we compute the multiplication in $\mathcal{A}$ approximately in
practice (see Subsection 6.1.4), its computational cost is much higher than
the multiplication in $\mathbb{C}$.

####  6.1.3 Minimization of the trace

In the case of $\mathcal{A}=\mathcal{B}(\mathcal{W})$, $p_{j}(x_{i})$ and
$p_{j}(x_{i})^{*}$ in the problem (9) do not always commute. Therefore, we
restrict the solution to the form
$p_{j}(x_{i})=\sum_{i=1}^{n}\phi(x_{i})c_{i}$ where each $c_{i}$ is a
Hilbert–Schmidt operator and minimize the trace of the objective function of
the problem (9) as follows:

$\inf_{\mathbf{c}_{j}\in F,\
\\{\sqrt{\mathbf{G}}\mathbf{c}_{j}\\}_{j=1}^{r}\mbox{\footnotesize:
ONS}}-\operatorname{tr}\bigg{(}\sum_{j=1}^{r}\mathbf{c}_{j}^{*}\mathbf{G}^{2}\mathbf{c}_{j}\bigg{)},$
(17)

where $F=\\{\mathbf{c}=[c_{1},\ldots,c_{n}]\in\mathcal{A}^{n}\mid\ c_{i}\mbox{
is a Hilbert--Schmidt operator for }i=1,\ldots,n\\}$. If
$\mathcal{A}=\mathbb{C}^{m\times m}$, i.e., $\mathcal{W}$ is a finite
dimensional space, then we solve the problem (17) by regarding $\mathbf{G}$ as
an $mn\times mn$ matrix and computing the eigenvalues and eigenvectors of
$\mathbf{G}$.

###### Proposition 6.8

Let $\mathcal{A}=\mathbb{C}^{m\times m}$. Let
$\lambda_{1},\ldots,\lambda_{r}\in\mathbb{C}$ and
$\mathbf{v}_{1},\ldots,\mathbf{v}_{r}\in\mathbb{C}^{mn}$ be the largest $r$
eigenvalues and the corresponding orthonormal eigenvectors of
$\mathbf{G}\in\mathbb{C}^{mn\times mn}$. Then,
$\mathbf{c}_{j}=[\mathbf{v}_{j},0,\ldots,0]\lambda_{j}^{-1/2}$ is a solution
of the problem (17).

Proof  Since the identity
$\sum_{j=1}^{r}\mathbf{c}_{j}^{*}\mathbf{G}^{2}\mathbf{c}_{j}=\sum_{j=1}^{r}(\sqrt{\mathbf{G}}\mathbf{c}_{j})^{*}\mathbf{G}(\sqrt{\mathbf{G}}\mathbf{c}_{j})$
holds, any solution $\mathbf{c}_{j}$ of the problem (17) satisfies
$\sqrt{\mathbf{G}}\mathbf{c}_{j}=\mathbf{v}_{j}u^{*}$ for a normalized vector
$u\in\mathbb{C}^{m}$. Thus, $p_{j}=\sum_{i=1}^{n}\phi(x_{i})c_{i,j}$, where
${c}_{i,j}$ is the $i$-th element of
$\lambda_{j}^{-1/2}[\mathbf{v}_{j},0,\ldots,0]$, is a solution of the problem.

If $\mathcal{W}$ is an infinite dimensional space, we rewrite the problem (17)
with the Hilbert–Schmidt norm as follows:

$\inf_{\mathbf{c_{j}}\in F,\
\\{\sqrt{\mathbf{G}}\mathbf{c}_{j}\\}_{j=1}^{r}\mbox{\footnotesize:
ONS}}-\sum_{j=1}^{r}\|\mathbf{G}\mathbf{c}_{j}\|^{2}_{F},$ (18)

where $\|\mathbf{c}\|_{F}^{2}=\sum_{i=1}^{n}\|c_{i}\|_{\operatorname{HS}}^{2}$
and $\|\cdot\|_{\operatorname{HS}}$ is the Hilbert–Schmidt norm for
Hilbert–Schmidt operators. Similar to Eq. (11), we rearrange the problem (18)
to the following problem by adding a penalty term:

$\inf_{\mathbf{c}\in
F}-\|\mathbf{G}\mathbf{c}\|^{2}_{F}+\lambda\big{|}\big{\|}\sqrt{\mathbf{Gc}}\big{\|}^{2}_{F}-1\big{|},$
(19)

where $\lambda$ is a real positive weight for the penalty term. Then, we can
apply the standard gradient descent method in Hilbert spaces to the problem in
$F$ (Smyrlis and Zisis, 2004) since $F$ is the Hilbert space equipped with the
Hilbert–Schmidt inner product. Similar to the case of Eq. (11), for $r>1$, let
$\mathbf{c}_{1}$ be a solution of the problem (19). Then, we solve the same
problem in the orthogonal complement of the space spanned by
$\\{\mathbf{c}_{1}\\}$ and set the solution of this problem as
$\mathbf{c}_{2}$. Then, we solve the same problem in the orthogonal complement
of the space spanned by $\\{\mathbf{c}_{1},\mathbf{c}_{2}\\}$ and repeat this
procedure to obtain solutions $\mathbf{c}_{1},\ldots\mathbf{c}_{r}$.

####  6.1.4 Numerical examples

##### Experiments with synthetic data

We applied the above PCA with $\mathcal{A}=L^{\infty}([0,1])$ to functional
data. We randomly generated three kinds of sample-sets from the following
functions of two variables on $[0,1]\times[0,1]$:

$\displaystyle y_{1}(s,t)=e^{10(s-t)},\quad y_{2}(s,t)=10st,\quad
y_{3}(s,t)=\cos(10(s-t)).$

Each sample-set $i$ is composed of 20 samples with random noise. We denote
these samples by $x_{1},\ldots,x_{60}$. The noise was randomly drawn from the
Gaussian distribution with mean $0$ and standard deviation $0.3$. Since
$L^{\infty}([0,1])$ is commutative, we applied the gradient descent proposed
in Subsection 6.1.2 to solve the problem (8). The parameters were set as
$\lambda=0.1$ and $\eta_{t}=0.01$. We set the $L^{\infty}([0,1])$-valued
positive definite kernel $k$ as
$(k(x_{i},x_{j}))(t)=\int_{0}^{1}\int_{0}^{1}(t-x_{i}(s_{1},s_{2}))(t-x_{j}(s_{1},s_{2}))ds_{1}ds_{2}$
(see Example 2.22.1). Since $(k(x_{i},x_{j}))(t)$ is a polynomial of $t$, all
the computations on $\mathcal{A}$ result in polynomials. Thus, the results are
obtained by keeping coefficients of the polynomials. Moreover, we set
$\mathbf{c}_{0}$ as the constant function $[1,\ldots,1]^{T}\in\mathcal{A}^{n}$
and computed $\mathbf{c}_{1},\mathbf{c}_{2},\ldots$ according to Eq. (14). For
comparison, we also vectorized the samples by discretizing $y_{i}$ at
$121=11\times 11$ points composed of 11 equally spaced points in $[0,1]$
($0,0.1,\ldots,1$) and applied the standard kernel PCA in the RKHS associated
with the Laplacian kernel on $\mathbb{R}^{121}$. The results are illustrated
in Figure 2. Since the samples are contaminated by the noise, the PCA in the
RKHS cannot separate three sample-sets. On the other hand, the
$L^{\infty}([0,1])$-valued weights of principal components obtained by the
proposed PCA in the RKHM reduce the information of the samples as functions.
As a result, it clearly separates three sample-sets.

Figure 3 shows the convergence of the proposed gradient descent. In this
example, we only compute the first principal components, hence $r$ is set as
$1$. For the objective function $f$ defined as
$f(\mathbf{c})=-\mathbf{c}^{*}\mathbf{G}^{2}\mathbf{c}+\lambda\mathbf{c}\mathbf{G}\mathbf{c}\mathbf{c}^{*}\mathbf{G}\mathbf{c}+\lambda\mathbf{c}^{*}\mathbf{G}\mathbf{c}$,
functions $f(\mathbf{c}_{t})\in L^{\infty}([0,1])$ for $t=0,\ldots,9$ are
illustrated. We can see $f(\mathbf{c}_{t+1})<f(\mathbf{c}_{t})$ and
$f(\mathbf{c}_{t})$ gradually approaches a certain function as $t$ grows.

Figure 2: The $L^{\infty}([0,1])$-valued first principal components obtained
by the proposed PCA in an RKHM (left) and the real-valued first and second
principal components obtained by the standard PCA in an RKHS (right) Figure 3:
The convergence of the function $f(\mathbf{c}_{t})$ along $t$.

##### Experiments with real-world data

To show the proposed PCA with RKHMs extracts the continuous dependencies of
samples on the principal axes as we insisted in Section 3, we conducted
experiments with climate data in Japan111available at
https://www.data.jma.go.jp/gmd/risk/obsdl/. The data is composed of the
maximum and minimum daily temperatures at 47 prefectures in Japan in 2020. The
original data is illustrated in Figure 4. The red line represents the
temperature at Hokkaido, the northernmost prefecture in Japan and the blue
line represents that at Okinawa, the southernmost prefecture in Japan. We
respectively fit the maximum and minimum temperatures at each location to the
Fourier series $a_{0}+\sum_{i=1}^{10}(a_{i}\cos(it)+b_{i}\sin(it))$. The
fitted functions $x_{1},\ldots,x_{47}\in C([0,366],\mathbb{R}^{2})$ are
illustrated in Figure 5. Then, we applied the PCA with the RKHM associated
with the $L^{\infty}([0,366])$-valued positive definite kernel
$(k(x,y))(t)=e^{-\|x(t)-y(t)\|_{2}^{2}}$. Let
$\mathcal{F}=\\{a_{0}+\sum_{i=1}^{10}(a_{i}\cos(it)+b_{i}\sin(it))\mid\
a_{i},b_{i}\in\mathbb{R}\\}\subseteq L^{2}([0,366])$. We project $k(x,y)$ onto
$\mathcal{F}$. Then, for $c,d\in\mathcal{F}$, $c+d\in\mathcal{F}$ is
satisfied, but $cd\in\mathcal{F}$ is not always satisfied. Thus, we
approximate $cd$ with $a_{0}+\sum_{i=1}^{N}(a_{i}\cos(it)+b_{i}\sin(it))$ for
$N\leq 10$ to restrict all the computations in $\mathcal{F}$ in practice.
Here, to remove high frequency components corresponding to noise and extract
essential information, we set $N=3$. Figure 6(a) shows the computed
$L^{\infty}([0,366])$-valued weights of the first principal axis in the RKHM,
which continuously depends on time. The red and blue lines correspond to
Hokkaido and Okinawa, respectively. We see these lines are well-separated from
other lines corresponding to other prefectures. For comparison, we also
applied the PCA in RKHSs to discrete time data. First, we respectively applied
the standard kernel PCA with RKHSs to the original temperature each day and
obtained real-valued weights of the first principal components. Here, we used
the complex-valued Gaussian kernel $\tilde{k}(x,y)=e^{-\|x-y\|_{2}^{2}}$.
Then, we connected the results and obtained Figure 6(b). Since the original
data is not smooth, the PCA amplifies the non-smoothness, which provides
meaningless results. Next, we respectively applied the standard kernel PCA
with the RKHS to the value of the fitted Fourier series each day and obtained
real-valued weights of the first principal components. Then, similar to the
case of Figure 6(b), we connected the results and obtained Figure 6(c). In
this case, the extracted features somewhat capture the continuous behaviors of
the temperatures. However, the PCA in the RKHS amplifies high frequency
components, which correspond to noise. Therefore, the result fails to separate
the temperatures of Hokkaido and Okinawa, whose behaviors are significantly
different as illustrated in Figure 4. On the other hand, the PCA in the RKHM
captures the feature of each sample as a function and removes nonessential
high frequency components, which results in separating functional data
properly.

Figure 4: Original climate data at 47 locations

Figure 5: Fitted Fourier series

(a) PCA with RKHMs for the fitted Fourier series

(b) PCA with RKHSs for the original data

(c) PCA with RKHSs for the fitted Fourier series

Figure 6: Principal components of PCA for climate data

###  6.2 Time-series data analysis

The problem of analyzing dynamical systems from data by using Perron–Frobenius
operators and their adjoints (called Koopman operators), which are linear
operators expressing the time evolution of dynamical systems, has recently
attracted attention in various fields (Budišić et al., 2012; Črnjarić-Žic et
al., 2020; Takeishi et al., 2017a, b; Lusch et al., 2018). And, several
methods for this problem using RKHSs have also been proposed (Kawahara, 2016;
Klus et al., 2020; Ishikawa et al., 2018; Hashimoto et al., 2020; Fujii &
Kawahara, 2019). In these methods, sequential data is supposed to be generated
from dynamical systems and is analyzed through Perron–Frobenius operators in
RKHSs. To analyze the time evolution of functional data, we generalize
Perron–Frobenius operators defined in RKHSs to those in RKHMs by using an
operator-valued positive definite kernel describing similarities between pairs
of functions.

##### Defining Perron–Frobenius operators in RKHMs

We consider the RKHM and vvRKHS associated with an operator-valued positive
definite kernel. VvRKHSs are associated with operator-valued kernels, and as
we stated in Lemma 4.10, those operator-valued kernels are special cases of
$C^{*}$-algebra-valued positive definite kernels. Here, we discuss the
advantage of RKHMs over vvRKHSs. Comparing with vvRKHSs, RKHMs have enough
representation power for preserving continuous behaviors of infinite
dimensional operator-valued kernels, while vvRKHSs are not sufficient for
preserving such behaviors. Let $\mathcal{W}$ be a Hilbert space, let
$k:\mathcal{X}\times\mathcal{X}\to\mathcal{B}(\mathcal{W})$ be an operator-
valued positive definite kernel on a data space $\mathcal{X}$, and let
$\mathcal{H}_{k}^{\operatorname{v}}$ be the vvRKHS associated with $k$. Since
the inner products in vvRKHSs have the form $\left\langle
w,k(x,y)h\right\rangle$ for $w,h\in\mathcal{W}$ and $x,y\in\mathcal{X}$, if
$\mathcal{W}$ is a $d$-dimensional space, putting $w$ as $d$ linearly
independent vectors in $\mathcal{W}$ reconstructs $k(x,y)$. However, if
$\mathcal{W}$ is an infinite dimensional space, we need infinitely many $w$ to
reconstruct $k(x,y)$, and we cannot recover the continuous behavior of the
operator $k(x,y)$ with finitely many $w$. For example, let
$\mathcal{X}=C(\Omega,\mathcal{Y})$ and $\mathcal{W}=L^{2}(\Omega)$ for a
compact measure space $\Omega$ and a topological space $\mathcal{Y}$. Let
$(k(x,y)w)(s)=\int_{t\in\Omega}\tilde{k}(x(s),y(t))w(t)dt$, where $\tilde{k}$
is a complex-valued positive definite kernel on $\mathcal{Y}$ (see Example
2.22.4). The operator $k(x,y)$ for functional data $x$ and $y$ describes the
continuous changes of similarities between function $x$ and $y$. However, the
estimation or prediction of the operator $k(x,y)$ in vvRKHSs fails to extract
the continuous behavior of the function $\tilde{k}(x(s),y(t))$ in the operator
$k(x,y)$ since vectors in vvRKHSs have the form $k(\cdot,y)w$ and we cannot
completely recover $k(x,y)$ with finitely many vectors in the vvRKHS. On the
other hand, RKHMs have enough information to recover $k(x,y)$ since it is just
the inner product between two vectors $\phi(x)$ and $\phi(y)$.

####  6.2.1 Perron–Frobenius operator in RKHSs

We briefly review the definition of the Perron-Frobenius operator on RKHS and
existing methods for analysis of time-series data through Perron–Frobenius
operators and construction of their estimations (Kawahara, 2016; Hashimoto et
al., 2020) . First, we define Perron–Frobenius operators in RKHSs. Let
$\\{x_{0},x_{1},\ldots\\}\subseteq\mathcal{X}$ be time-series data. We assume
it is generated from the following deterministic dynamical system:

$x_{i+1}=f(x_{i}),$ (20)

where $f:\mathcal{X}\to\mathcal{X}$ is a map. By embedding $x_{i}$ and
$f(x_{i})$ in an RKHS $\mathcal{H}_{\tilde{k}}$ associated with a positive
definite kernel $\tilde{k}$ and the feature map $\tilde{\phi}$, dynamical
system (20) in $\mathcal{X}$ is transformed into that in the RKHS as

$\tilde{\phi}(x_{i+1})=\tilde{\phi}(f(x_{i})).$

The Perron–Frobenius operator $\tilde{K}$ in the RKHS is defined as a linear
operator on $\mathcal{H}_{\tilde{k}}$ satisfying

$\tilde{K}\tilde{\phi}(x):=\tilde{\phi}(f(x))$

for $x\in\mathcal{X}$. If $\\{\tilde{\phi}(x)\mid\ x\in\mathcal{X}\\}$ is
linearly independent, $\tilde{K}$ is well-defined as a linear map in the RKHS.
For example, if $\tilde{k}$ is a universal kernel (Sriperumbudur et al., 2011)
such as the Gaussian or Laplacian kernel on $\mathcal{X}=\mathbb{R}^{d}$,
$\\{\tilde{\phi}(x)\mid\ x\in\mathcal{X}\\}$ is linearly independent.

By considering eigenvalues and the corresponding eigenvectors of $\tilde{K}$,
we can understand the long-time behavior of the dynamical system. For example,
let $v_{1},\ldots,v_{m}$ be the eigenvectors with respect to eigenvalue 1 of
$\tilde{K}$. We project the vector $\tilde{\phi}(x_{0})$ onto the subspace
spanned by $v_{1},\dots,v_{m}$. We denote the projected vector by $v$. Then,
for $\alpha=1,2,\ldots$, we have

$\tilde{\phi}(x_{\alpha})=\tilde{K}^{\alpha}(v+v^{\perp})=v+\tilde{K}^{\alpha}v^{\perp},$

where $v^{\perp}=\tilde{\phi}(x_{0})-v$. Therefore, by calculating a pre-image
of $v$, we can extract the time-invariant component of the dynamical system
with the initial value $x_{0}$.

For practical uses of the above discussion, we construct an estimation of
$\tilde{K}$ only with observed data
$\\{x_{0},x_{1},\ldots\\}\subseteq\mathcal{X}$ as follows: We project
$\tilde{K}$ onto the finite dimensional subspace spanned by
$\\{\tilde{\phi}(x_{0}),\ldots,\tilde{\phi}(x_{T-1})\\}$. Let
$\tilde{W}_{T}:=[\tilde{\phi}(x_{0}),\ldots,\tilde{\phi}(x_{T-1})]$ and
$\tilde{W}_{T}=\tilde{Q}_{T}\tilde{\mathbf{R}}_{T}$ be the QR decomposition of
$\tilde{W}_{T}$ in the RKHS. Then, the Perron–Frobenius operator $\tilde{K}$
is estimated by projecting $\tilde{K}$ onto the space spanned by
$\\{\tilde{\phi}(x_{0}),\ldots,\tilde{\phi}(x_{T-1})\\}$. Since
$\tilde{K}\tilde{\phi}(x_{i}):=\tilde{\phi}(f(x_{i}))=\tilde{\phi}(x_{i+1})$
holds, we construct an estimation $\tilde{\mathbf{K}}_{T}$ of $\tilde{K}$ as
follows:

$\displaystyle\tilde{\mathbf{K}}_{T}:$
$\displaystyle=\tilde{Q}_{T}^{*}\tilde{K}\tilde{Q}_{T}=\tilde{Q}_{T}^{*}\tilde{K}\tilde{W}_{T}\tilde{\mathbf{R}}_{T}^{-1}=\tilde{Q}_{T}^{*}[\tilde{\phi}(x_{1}),\ldots,\tilde{\phi}(x_{T})]\tilde{\mathbf{R}}_{T}^{-1},$

which can be computed only with observed data.

####  6.2.2 Perron–Frobenius operator in RKHMs

Existing analyses (Kawahara, 2016; Hashimoto et al., 2020) of time-series data
with Perron–Frobenius operators are addressed only in RKHSs. In the remaining
parts of this section, we generalize the existing analyses to RKHM to extract
continuous behaviors of functional data. We consider the case where time-
series is functional data. Let $\Omega$ be a compact measure space,
$\mathcal{Y}$ be a topological space, $\mathcal{X}=C(\Omega,\mathcal{Y})$,
$\mathcal{A}=\mathcal{B}(L^{2}(\Omega))$, and
$\\{x_{0},x_{1},\ldots\\}\subseteq\mathcal{X}$ be functional time-series data.
Let $k:\mathcal{X}\times\mathcal{X}\to\mathcal{A}$ be defined as
$(k(x,y)w)(s)=\int_{t\in\Omega}\tilde{k}(x(s),y(t))w(t)dt$, where
$\tilde{k}:\mathcal{Y}\times\mathcal{Y}\to\mathbb{C}$ is a complex-valued
positive definite kernel (see Example 2.22.4 and the last paragraph of Section
3). The operator $k(x,y)$ is the integral operator whose integral kernel is
$\tilde{k}(x(s),y(t))$. We define a Perron–Frobenius operator in the RKHM
$\mathcal{M}_{k}$ associated with the above kernel $k$ as an
$\mathcal{A}$-linear operator satisfying

${K}{\phi}(x)={\phi}(f(x))$

for $x\in\mathcal{X}$. We assume $K$ is well-defined on a dense subset of
$\mathcal{M}_{k}$. Then, for $\alpha,\beta=1,2,\ldots$, we have

$k(x_{\alpha},x_{\beta})=\left\langle\phi(x_{\alpha}),\phi(x_{\beta})\right\rangle_{\mathcal{M}_{k}}=\big{\langle}K^{\alpha}\phi(x_{0}),K^{\beta}\phi(x_{0})\big{\rangle}_{\mathcal{M}_{k}}.$

Therefore, by estimating $K$ in the RKHM $\mathcal{M}_{k}$, we can extract the
similarity between arbitrary points of functions $x_{\alpha}$ and $x_{\beta}$.
Moreover, the eigenvalues and eigenvectors of $K$ provide us a decomposition
of the similarity $k(x_{\alpha},x_{\beta})$ into a time-invariant term and
time-dependent term. Since $K$ is a linear operator on a Banach space
$\mathcal{M}_{k}$, eigenvalues and eigenvectors of $K$ are available. Let
$v_{1},\ldots,v_{m}\in\mathcal{M}_{k}$ be the eigenvectors with respect to
eigenvalue $1$ of $K$. We project the vector $\phi(x_{0})$ onto the submodule
spanned by $v_{1},\ldots,v_{m}$, which is denoted by $\mathcal{V}$. Let
$\\{q_{1},\ldots,q_{m}\\}\subseteq\mathcal{M}_{k}$ be an orthonormal basis of
$\mathcal{V}$ and let $v=\sum_{i=1}^{m}q_{i}\left\langle
q_{i},\phi(x_{0})\right\rangle_{\mathcal{M}_{k}}$. Then, we have

$k(x_{\alpha},x_{\beta})=\big{\langle}K^{\alpha}(v+v^{\perp}),K^{\beta}(v+v^{\perp})\big{\rangle}_{\mathcal{M}_{k}}=\left\langle
v,v\right\rangle_{\mathcal{M}_{k}}+r({\alpha},\beta),$ (21)

where $v^{\perp}=\phi(x_{0})-v$ and $r(\alpha,\beta)=\left\langle
K^{\alpha}v,K^{\beta}v^{\perp}\right\rangle_{\mathcal{M}_{k}}+\left\langle
K^{\alpha}v^{\perp},K^{\beta}v\right\rangle_{\mathcal{M}_{k}}+\left\langle
K^{\alpha}v^{\perp},K^{\beta}v^{\perp}\right\rangle_{\mathcal{M}_{k}}$.
Therefore, the term $\left\langle v,v\right\rangle_{\mathcal{M}_{k}}$ provides
us with the information about time-invariant similarities.

###### Remark 6.9

We can also consider the vvRKHS $\mathcal{H}_{k}^{\operatorname{v}}$ with
respect to the operator-valued kernel $k$. Here, we discuss the difference
between the case of vvRKHS and RKHM. The Perron–Frobenius operator
$K^{\operatorname{v}}$ in a vvRKHS $\mathcal{H}_{k}^{\operatorname{v}}$ (Fujii
& Kawahara, 2019) is defined as a linear operator satisfying

$K^{\operatorname{v}}\phi(x)w=\phi(f(x))w$

for $x\in\mathcal{X}$ and $w\in\mathcal{W}$. However, with finitely many
vectors in $\mathcal{H}_{k}^{\operatorname{v}}$, we can only recover an
projected operator $UU^{*}k(x_{\alpha},x_{\beta})UU^{*}$, where
$N\in\mathbb{N}$, $U=[u_{1},\ldots,u_{N}]$, and $\\{u_{1},\ldots,u_{N}\\}$ is
an orthonormal system on $\mathcal{W}$ as follows:

$U^{*}k(x_{\alpha},x_{\beta})U=\big{[}\left\langle\phi(x_{s})u_{i},\phi(x_{t})u_{j}\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}\big{]}_{i,j}=\big{[}\big{\langle}(K^{\operatorname{v}})^{\alpha}\phi(x_{0})u_{i},(K^{\operatorname{v}})^{\beta}\phi(x_{0})u_{j}\big{\rangle}_{\mathcal{H}_{k}^{\operatorname{v}}}\big{]}_{i,j}.$
(22)

Furthermore, let $v_{1},\ldots,v_{m}\in\mathcal{M}_{k}$ be the eigenvectors
with respect to eigenvalue $1$ of $K^{\operatorname{v}}$. Let
$\\{q_{1},\ldots,q_{m}\\}\subseteq\mathcal{H}_{k}^{\operatorname{v}}$ be an
orthonormal basis of the subspace spanned by $v_{1},\ldots,v_{m}$ and let
$\tilde{v}_{j}=\sum_{i=1}^{m}q_{i}\left\langle
q_{i},\phi(x_{0})u_{j}\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}$.
Then, we have

$U^{*}k(x_{\alpha},x_{\beta})U=\big{[}\langle(K^{\operatorname{v}})^{\alpha}(\tilde{v}_{i}+\tilde{v}_{i}^{\perp}),(K^{\operatorname{v}})^{\beta}(\tilde{v}_{j}+\tilde{v}_{j}^{\perp})\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}\big{]}_{i,j}=[\left\langle\tilde{v}_{i},\tilde{v}_{j}\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}]_{i,j}+\tilde{r}(\alpha,\beta),$
(23)

where $\tilde{v}_{i}^{\perp}=\phi(x_{0})u_{i}-\tilde{v}_{i}$ and
$\tilde{r}(\alpha,\beta)=[\langle(K^{\operatorname{v}})^{\alpha}\tilde{v}_{i},(K^{\operatorname{v}})^{\beta}\tilde{v}_{j}^{\perp}\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}+\langle(K^{\operatorname{v}})^{\alpha}\tilde{v}_{i}^{\perp},(K^{\operatorname{v}})^{\beta}\tilde{v}_{j}\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}+\langle(K^{\operatorname{v}})^{\alpha}\tilde{v}_{i}^{\perp},(K^{\operatorname{v}})^{\beta}\tilde{v}_{j}^{\perp}\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}]_{i,j}$.
Therefore, with vvRKHSs, we cannot recover the continuous behavior of the
operator $k(x,y)$ which encodes similarities between functions $x$ and $y$.

####  6.2.3 Estimation of Perron–Frobenius operators in RKHMs

In practice, we only have time-series data but do not know the underlying
dynamical system and its Perron–Frobenius operator in an RKHM. Therefore, we
consider estimating the Perron–Frobenius operator only with the data. To do
so, we generalize the Gram–Schmidt orthonormalization algorithm to Hilbert
$C^{*}$-modules to apply the QR decomposition and project Perron–Frobenius
operators onto the submodule spanned by
$\\{\phi(x_{0}),\ldots,\phi(x_{T-1})\\}$. The Gram–Schmidt orthonormalization
in Hilbert modules is theoretically investigated by Cnops (1992). Here, we
develop a practical method for our settings. Then, we can apply the
decomposition (21), proposed in Subsection 6.2.2, of the estimated operator
regarding eigenvectors. Since we are considering the RKHM associated with the
integral operator-valued positive definite kernel defined in the first part of
Subsection 6.2.2, we assume $\mathcal{A}=\mathcal{B}(\mathcal{W})$ and we
denote by $\mathcal{M}$ a Hilbert $C^{*}$-module over $\mathcal{A}$ throughout
this subsection. Note that integral operators are compact.

We first develop a normalization method for Hilbert $C^{*}$-modules. In
$C^{*}$-algebras, nonzero elements are not always invertible, which is the
main difficulty of the normalization in Hilbert $C^{*}$-modules. However, by
carefully applying the definition of normalized (see Definition 2.17), we can
construct a normalization method.

######  Proposition 6.10 (Normalization)

Let $\epsilon\geq 0$ and let $\hat{q}\in\mathcal{M}$ satisfy
$\|\hat{q}\|_{\mathcal{M}}>\epsilon$. Assume
$\langle\hat{q},\hat{q}\rangle_{\mathcal{M}}$ is compact. Then, there exists
$\hat{b}\in\mathcal{A}$ such that $\|\hat{b}\|_{\mathcal{A}}<1/\epsilon$ and
$q:=\hat{q}\hat{b}$ is normalized. In addition, there exists $b\in\mathcal{A}$
such that $\|\hat{q}-qb\|_{\mathcal{M}}\leq\epsilon$.

Proof  Let $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq 0$ be the eigenvelues of
the compact operator $\left\langle\hat{q},\hat{q}\right\rangle_{\mathcal{M}}$,
and $m^{\prime}:=\max\\{j\mid\ \lambda_{j}>\epsilon^{2}\\}$. Since
$\langle\hat{q},\hat{q}\rangle_{\mathcal{M}}$ is positive and compact, it
admits the spectral decomposition
$\langle\hat{q},\hat{q}\rangle_{\mathcal{M}}=\sum_{i=1}^{\infty}\lambda_{i}v_{i}v_{i}^{*}$,
where $v_{i}$ is the orthonormal eigenvector with respect to $\lambda_{i}$.
Also, since $\lambda_{1}=\|\hat{q}\|_{\mathcal{M}}^{2}>\epsilon^{2}$, we have
$m^{\prime}\geq 1$. Let
$\hat{b}=\sum_{i=1}^{m^{\prime}}1/\sqrt{\lambda_{i}}v_{i}v_{i}^{*}$. By the
definition of $\hat{b}$,
$\|\hat{b}\|_{\mathcal{A}}=1/\sqrt{\lambda_{m^{\prime}}}<1/\epsilon$ holds.
Also, we have

$\displaystyle\langle\hat{q}\hat{b},\hat{q}\hat{b}\rangle_{\mathcal{M}}$
$\displaystyle=\hat{b}^{*}\left\langle\hat{q},\hat{q}\right\rangle_{\mathcal{M}}\hat{b}=\sum_{i=1}^{m^{\prime}}\frac{1}{\sqrt{\lambda_{i}}}v_{i}v_{i}^{*}\sum_{i=1}^{\infty}\lambda_{i}v_{i}v_{i}^{*}\sum_{i=1}^{m^{\prime}}\frac{1}{\sqrt{\lambda_{i}}}v_{i}v_{i}^{*}=\sum_{i=1}^{m^{\prime}}v_{i}v_{i}^{*}.$

Thus, $\langle\hat{q}\hat{b},\hat{q}\hat{b}\rangle_{\mathcal{M}}$ is a nonzero
orthogonal projection.

In addition, let $b=\sum_{i=1}^{m^{\prime}}\sqrt{\lambda_{i}}v_{i}v_{i}^{*}$.
Since $\hat{b}b=\sum_{i=1}^{m^{\prime}}v_{i}v_{i}^{*}$, the identity
$\langle\hat{q},\hat{q}\hat{b}b\rangle=\langle\hat{q}\hat{b}b,\hat{q}\hat{b}b\rangle$
holds, and we obtain

$\displaystyle\langle\hat{q}-qb,\hat{q}-qb\rangle_{\mathcal{M}}$
$\displaystyle=\langle\hat{q}-\hat{q}\hat{b}b,\hat{q}-\hat{q}\hat{b}b\rangle_{\mathcal{M}}=\langle\hat{q},\hat{q}\rangle-\langle\hat{q}\hat{b}b,\hat{q}\hat{b}b\rangle_{\mathcal{M}}$
$\displaystyle=\sum_{i=1}^{\infty}{\lambda_{i}}v_{i}v_{i}^{*}-\sum_{i=1}^{m^{\prime}}{\lambda_{i}}v_{i}v_{i}^{*}=\sum_{i=m^{\prime}+1}^{\infty}{\lambda_{i}}v_{i}v_{i}^{*}.$

Thus,
$\|\hat{q}-q\hat{b}\|_{\mathcal{M}}=\sqrt{\lambda_{m^{\prime}+1}}\leq\epsilon$
holds, which completes the proof of the proposition.
Proposition 6.10 and its proof provide a concrete procedure to obtain
normalized vectors in $\mathcal{M}$. This enables us to compute an orthonormal
basis practically by applying Gram-Schmidt orthonormalization with respect to
$\mathcal{A}$-valued inner product.

######  Proposition 6.11 (Gram-Schmidt orthonormalization)

Let $\\{w_{i}\\}_{i=1}^{\infty}$ be a sequence in $\mathcal{M}$. Assume
$\left\langle w_{i},w_{j}\right\rangle_{\mathcal{M}}$ is compact for any
$i,j=1,2,\ldots$. Consider the following scheme for $i=1,2,\ldots$ and
$\epsilon\geq 0$:

$\displaystyle\hat{q}_{j}$
$\displaystyle=w_{j}-\sum_{i=1}^{j-1}q_{i}\left\langle
q_{i},w_{j}\right\rangle_{\mathcal{M}},\quad
q_{j}=\hat{q}_{j}\hat{b}_{j}\quad\mbox{if
}\;\|\hat{q}_{j}\|_{\mathcal{M}}>\epsilon,$ (24) $\displaystyle q_{j}$
$\displaystyle=0\quad\mbox{o.w.},$

where $\hat{b}_{j}$ is defined as $\hat{b}$ in Proposition 6.10 by setting
$\hat{q}=\hat{q}_{j}$. Then, $\\{q_{j}\\}_{j=1}^{\infty}$ is an orthonormal
basis in $\mathcal{M}$ such that any $w_{j}$ is contained in the
$\epsilon$-neighborhood of the module spanned by $\\{q_{j}\\}_{j=1}^{\infty}$.

###### Remark 6.12

We give some remarks about the role of $\epsilon$ in Propositions 6.10. The
vector $\hat{q}_{i}$ can always be reconstructed by $w_{i}$ only when
$\epsilon=0$. This is because the information of the spectrum of
$\left\langle\hat{q}_{i},\hat{q}_{i}\right\rangle_{\mathcal{M}}$ may be lost
if $\epsilon>0$. However, if $\epsilon$ is sufficiently small, we can
reconstruct $\hat{q}_{i}$ with a small error. On the other hand, the norm of
$\hat{b}_{i}$ can be large if $\epsilon$ is small, and the computation of
$\\{q_{i}\\}_{i=1}^{\infty}$ can become numerically unstable. This corresponds
to the trade-off between the theoretical accuracy and numerical stability.

To prove Proposition 6.11, we first prove the following lemmas.

###### Lemma 6.13

For $c\in\mathcal{A}$ and $v\in\mathcal{M}$, if $\left\langle
v,v\right\rangle_{\mathcal{M}}c=\left\langle v,v\right\rangle_{\mathcal{M}}$,
then $vc=v$ holds.

Proof  If $\left\langle v,v\right\rangle_{\mathcal{M}}c=\left\langle
v,v\right\rangle_{\mathcal{M}}$, then $c^{*}\left\langle
v,v\right\rangle_{\mathcal{M}}=\left\langle v,v\right\rangle_{\mathcal{M}}$
and we have

$\displaystyle\left\langle
vc-v,vc-v\right\rangle_{\mathcal{M}}=c^{*}\left\langle
v,v\right\rangle_{\mathcal{M}}c-c^{*}\left\langle
v,v\right\rangle_{\mathcal{M}}-\left\langle
v,v\right\rangle_{\mathcal{M}}c+\left\langle
v,v\right\rangle_{\mathcal{M}}=0,$

which implies $vc=v$.

###### Lemma 6.14

If $q\in\mathcal{M}$ is normalized, then $q\left\langle
q,q\right\rangle_{\mathcal{M}}=q$ holds.

Proof  Since $\left\langle q,q\right\rangle_{\mathcal{M}}$ is a projection,
$\left\langle q,q\right\rangle_{\mathcal{M}}\left\langle
q,q\right\rangle_{\mathcal{M}}=\left\langle q,q\right\rangle_{\mathcal{M}}$
holds. Therefore, letting $c=\left\langle q,q\right\rangle_{\mathcal{M}}$ and
$v=q$ in Lemma 6.13 completes the proof of the lemma.

Proof of Proposition 6.11 By Proposition 6.10, $q_{j}$ is normalized, and for
$\epsilon\geq 0$, there exists $b_{j}\in\mathcal{A}$ such that
$\|\hat{q}_{j}-q_{j}b_{j}\|_{\mathcal{M}}\leq\epsilon$. Therefore, by the
definition of $\hat{q}_{j}$, $\|w_{j}-v_{j}\|_{\mathcal{M}}\leq\epsilon$
holds, where $v_{j}$ is a vector in the module spanned by
$\\{q_{j}\\}_{j=0}^{\infty}$ which is defined as
$v_{j}=\sum_{i=1}^{j-1}q_{i}\left\langle
q_{i},w_{j}\right\rangle_{\mathcal{M}}-q_{j}b_{j}$. This means that the
$\epsilon$-neighborhood of the space spanned by $\\{q_{j}\\}_{j=1}^{\infty}$
contains $\\{w_{j}\\}_{j=1}^{\infty}$. Next, we show the orthogonality of
$\\{q_{j}\\}_{j=1}^{\infty}$. Assume $q_{1},\ldots,q_{j-1}$ are orthogonal to
each other. For $i<j$, the following identities are deduced by Lemma 6.14:

$\displaystyle\left\langle q_{j},q_{i}\right\rangle_{\mathcal{M}}$
$\displaystyle=\hat{b}_{t}^{*}\left\langle\hat{q}_{j},q_{i}\right\rangle_{\mathcal{M}}=\hat{b}_{j}^{*}\bigg{\langle}w_{j}-\sum_{l=1}^{j-1}q_{l}\left\langle
q_{l},w_{j}\right\rangle,q_{i}\bigg{\rangle}_{\mathcal{M}}$
$\displaystyle=\hat{b}_{j}^{*}\left(\left\langle
w_{j},q_{i}\right\rangle_{\mathcal{M}}-\left\langle q_{i}\left\langle
q_{i},w_{j}\right\rangle_{\mathcal{M}},q_{i}\right\rangle\right)=\hat{b}_{j}^{*}\left(\left\langle
w_{j},q_{i}\right\rangle_{\mathcal{M}}-\left\langle
w_{j},q_{i}\right\rangle_{\mathcal{M}}\right)=0.$

Therefore, $q_{1},\ldots,q_{j}$ are also orthogonal to each other, which
completes the proof of the proposition.

In practical computations, the scheme (24) should be represented with
matrices. For this purpose, we derive the following QR decomposition from
Proposition 6.11. This is a generalization of the QR decomposition in Hilbert
spaces.

######  Corollary 6.15 (QR decomposition)

For $n\in\mathbb{N}$, let $W:=[w_{1},\ldots,w_{n}]$ and
$Q:=[q_{1},\ldots,q_{n}]$. Let $\epsilon\geq 0$. Then, there exist
$\mathbf{R},\mathbf{R}_{\operatorname{inv}}\in\mathcal{A}^{n\times n}$ that
satisfy

$Q=W\mathbf{R}_{\operatorname{inv}},\quad\|W-Q\mathbf{R}\|\leq\epsilon.$ (25)

Here, $\|W\|$ for a $\mathcal{A}$-linear map $W:\mathcal{A}^{n}\to\mathcal{M}$
is defined as $\|W\|:=\sup_{\|v\|_{\mathcal{A}^{n}}=1}\|Wv\|_{\mathcal{M}}$.

Proof  Let $\mathbf{R}=[r_{i,j}]_{i,j}$ be an $n\times n$ $\mathcal{A}$-valued
matrix. Here, $r_{i,j}$ is defined by $r_{i,j}=\left\langle
q_{i},w_{j}\right\rangle_{\mathcal{M}}\in\mathcal{A}$ for $i<j$, $r_{i,j}=0$
for $i>j$, and $r_{j,j}=b_{j}$, where $b_{j}$ is defined as $b$ in Proposition
6.10 by setting $\hat{q}=\hat{q}_{j}$. In addition, let
$\hat{\mathbf{B}}=\operatorname{diag}\\{\hat{b}_{1},\ldots,\hat{b}_{n}\\}$,
$\mathbf{B}=\operatorname{diag}\\{{b}_{1},\ldots,{b}_{n}\\}$, and
$\mathbf{R}_{\operatorname{inv}}=\mathbf{\hat{B}}(I+(\mathbf{R}-\mathbf{B})\mathbf{\hat{B}})^{-1}$
be $n\times n$ $\mathcal{A}$-valued matrices. The equality
$Q=W\mathbf{R}_{\operatorname{inv}}$ is derived directly from scheme (24). In
addition, by the scheme (24), for $t=1,\ldots,n$, we have

$\displaystyle w_{j}$ $\displaystyle=\sum_{i=1}^{j-1}q_{i}\left\langle
q_{i},w_{j}\right\rangle_{\mathcal{M}}+\hat{q}_{j}=\sum_{i=1}^{j-1}q_{i}\left\langle
q_{i},w_{j}\right\rangle_{\mathcal{M}}+q_{j}b_{j}+\hat{q}_{j}-q_{j}b_{j}=Q\mathbf{r}_{j}+\hat{q}_{j}-q_{j}b_{j},$

where $\mathbf{r}_{j}\in\mathcal{A}^{n}$ is the $i$-th column of $\mathbf{R}$.
Therefore, by Proposition 6.10,
$\|w_{j}-Q\mathbf{r}_{j}\|_{\mathcal{M}}=\|\hat{q}_{j}-q_{j}b_{j}\|_{\mathcal{M}}\leq\epsilon$
holds for $j=1,\ldots,n$, which implies $\|W-Q\mathbf{R}\|\leq\epsilon$.
We call the decomposition (25) as the QR decomposition in Hilbert
$C^{*}$-modules. Although we are handling vectors in $\mathcal{M}$, by
applying the QR decomposition, we only have to compute
$\mathbf{R}_{\operatorname{inv}}$ and $\mathbf{R}$.

We now consider estimating the Perron–Frobenius operator $K$ with observed
time-series data $\\{x_{0},x_{1},\ldots\\}$. Let
${W}_{T}=[{\phi}(x_{0}),\ldots,{\phi}(x_{T-1})]$. We are considering an
integral operator-valued positive definite kernel (see the first part of
Subsection 6.2.2 and the last paragraph in Section 3). Since integral
operators are compact, $W_{T}$ satisfies the assumption in Corollary 6.11.
Thus, let ${W}_{T}{\mathbf{R}}_{\operatorname{inv},T}={Q}_{T}$ be the QR
decomposition (25) of ${W}_{T}$ in the RKHM $\mathcal{M}_{k}$. The
Perron–Frobenius operator $K$ is estimated by projecting ${K}$ onto the module
spanned by $\\{{\phi}(x_{0}),\ldots,\phi(x_{T-1})\\}$. We define
${\mathbf{K}}_{T}$ as the estimation of $K$. Since
${K}{\phi}(x_{i})={\phi}(f(x_{i}))={\phi}(x_{i+1})$ hold, ${\mathbf{K}}_{T}$
can be computed only with observed data as follows:

$\displaystyle{\mathbf{K}}_{T}$
$\displaystyle={Q}_{T}^{*}{K}{Q}_{T}={Q}_{T}^{*}{K}{W}_{T}{\mathbf{R}}_{\operatorname{inv},T}={Q}_{T}^{*}[{\phi}(x_{1}),\ldots,{\phi}(x_{T})]{\mathbf{R}}_{\operatorname{inv},T}.$

###### Remark 6.16

In practical computations, we only need to keep the integral kernels to
implement the Gram–Schmidt orthonormalization algorithm and estimate
Perron–Frobenius operators in the RKHM associated with the integral operator-
valued kernel $k$. Therefore, we can directly access integral kernel functions
of operators, which is not achieved by vvRKHS as we stated in Remark 4.13.
Indeed, the operations required for estimating Perron–Frobenius operators are
explicitly computed as follows: Let $c,d\in\mathcal{B}(L^{2}(\Omega))$ be
integral operators whose integral kernels are $f(s,t)$ and $g(s,t)$. Then, the
integral kernels of the operator $c+d$ and $cd$ are $f(s,t)+g(s,t)$ and
$\int_{r\in\Omega}f(s,r)g(r,t)dr$, respectively. And that of $c^{*}$ is
$f(t,s)$. Moreover, if $c$ is positive, let $c_{\epsilon}^{+}$ be
$\sum_{\lambda_{i}>\epsilon}1/\sqrt{\lambda_{i}}v_{i}{v_{i}}^{*}$, where
$\lambda_{i}$ are eigenvalues of the compact positive operator $c$ and $v_{i}$
are corresponding orthonormal eigenvectors. Then, the integral kernel of the
operator $c_{\epsilon}^{+}$ is
$\sum_{\lambda_{i}>\epsilon}1/\sqrt{\lambda_{i}}v_{i}(s)\overline{v_{i}(t)}$.

####  6.2.4 Numerical examples

To show the proposed analysis with RKHMs captures continuous changes of values
of kernels along functional data as we insisted in Section 3, we conducted
experiments with river flow data of the Thames River in London222available at
https://nrfa.ceh.ac.uk/data/search. The data is composed of daily flow at 10
stations. We used the data for 51 days beginning from January first, 2018. We
regard every daily flow as a function of the ratio of the distance from the
most downstream station and fit it to a polynomial of degree 5 to obtain time
series $x_{0},\ldots,x_{50}\in C([0,1],\mathbb{R})$. Then, we estimated the
Perron–Frobenius operator which describes the time evolution of the series
$x_{0},\ldots,x_{50}$ in the RKHM associated with the
$\mathcal{B}(L^{2}([0,1]))$-valued positive definite kernel $k(x,y)$ defined
as the integral operator whose integral kernel is
$\tilde{k}(s,t)=e^{-|x(s)-y(t)|^{2}}$ for $x,y\in C([0,1],\mathbb{R})$. In
this case, $T=50$. As we noted in Remark 6.16, all the computations in
$\mathcal{A}=\mathcal{B}(L^{2}([0,1]))$ are implemented by keeping integral
kernels of operators. Let $\mathcal{F}$ be the set of polynomials of the form
$x_{i}(s,t)=\sum_{j,l=0}^{5}\eta_{j,l}s^{j}t^{l}$, where
$\eta_{j,l}\in\mathbb{R}$. We project $\tilde{k}$ onto $\mathcal{F}$. Then,
for $c,d\in\mathcal{F}$, $c+d\in\mathcal{F}$ is satisfied, but
$cd\in\mathcal{F}$ is not always satisfied. Thus, we project $cd$ onto
$\mathcal{F}$ to restrict all the computations in $\mathcal{F}$ in practice.
We computed the time-invariant term $\left\langle
v,v\right\rangle_{\mathcal{M}_{k}}$ in Eq. (22). Regarding the computation of
eigenvectors with respect to the eigenvalue $1$, we consider the following
minimization problem for the estimated Perron–Frobenius operator
$\mathbf{K}_{T}$:

$\inf_{\mathbf{v}\in\mathcal{A}^{T}}|\mathbf{K}_{T}\mathbf{v}-\mathbf{v}|_{\mathcal{A}^{T}}^{2}-\lambda|\mathbf{v}|_{\mathcal{A}^{T}}^{2}.$
(26)

Here, $-\lambda|\mathbf{v}|_{\mathcal{A}^{T}}^{2}$ is a penalty term to keep
$\mathbf{v}$ not going to $0$. Since the objective function of the problem
(26) is represented as
$\mathbf{v}^{*}(\mathbf{K}_{T}^{*}\mathbf{K}_{T}-\mathbf{K}_{T}^{*}-\mathbf{K}_{T}+(1-\lambda)\mathbf{I})\mathbf{v}$,
where $\mathbf{I}$ is the identity operator on $\mathcal{A}^{T}$, we apply the
gradient descent on $\mathcal{A}^{T}$ (see Remark 6.6). Figure 7(a) shows the
heat map representing the integral kernel of $\left\langle
v,v\right\rangle_{\mathcal{M}_{k}}$.

For comparison, we also applied the similar analysis in a vvRKHS. We computed
the time-invariant term
$[\left\langle\tilde{v}_{i},\tilde{v}_{j}\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}]_{i,j}$
in Eq. (23) by setting $u_{i}$ as orthonormal polynomials of the form
$u_{i}(s)=\sum_{j=1}^{5}\eta_{j}s^{j}$, where $\eta_{j}\in\mathbb{R}$. Let
$c_{\operatorname{inv}}=[\left\langle\tilde{v}_{i},\tilde{v}_{j}\right\rangle_{\mathcal{H}_{k}^{\operatorname{v}}}]_{i,j}$.