#  $C^{*}$-algebra Net: A New Approach Generalizing
Neural Network Parameters to $C^{*}$-algebra

Yuka Hashimoto  Zhao Wang  Tomoko Matsui

###### Abstract

We propose a new framework that generalizes the parameters of neural network
models to $C^{*}$-algebra-valued ones. $C^{*}$-algebra is a generalization of
the space of complex numbers. A typical example is the space of continuous
functions on a compact space. This generalization enables us to combine
multiple models continuously and use tools for functions such as regression
and integration. Consequently, we can learn features of data efficiently and
adapt the models to problems continuously. We apply our framework to practical
problems such as density estimation and few-shot learning and show that our
framework enables us to learn features of data even with a limited number of
samples. Our new framework highlights the potential possibility of applying
the theory of $C^{*}$-algebra to general neural network models.

$C^{*}$-algebra, Hilbert $C^{*}$-module, Density estimation, Few-shot learning

##  1 Introduction

Continuation of neural network models has been discussed and successfully
applied to practical problems and theoretical analyses. Chen et al. (2018)
proposed to regard the transformation of input variables as a continuous
dynamical system, which is called the neural ODE. While the neural ODE focus
on the continuation of layers (vertical direction), continuation of units
(horizontal direction) is also discussed. By regarding the action of a weight
matrix to variables as integrations and using an integral representation of
the model, we can use the theory of harmonic analysis to analyze the model
theoretically (Candès, 1999; Sonoda & Murata, 2017; Sonoda et al., 2021). In
these frameworks, we regard the parameters of a classical model being obtained
by a discretization of functions, which enables us to use tools for functions
such as derivative and integration for practical applications and theoretical
analyses of the model.

In this paper, we propose a brand new framework that generalizes the
parameters (weights) of models. Unlike previous works regarding the
continuation of neural network models, we do not consider the continuation of
the parameters of a single classical model to functions. Instead, we
generalize each $\mathbb{R}$-valued parameter to a $C^{*}$-algebra-valued one,
which corresponds to an aggregation of multiple classical models and a
continuation of the parameters of the multiple models to functions.
$C^{*}$-algebra is a generalization of the space of complex numbers. Typical
examples are the space of continuous functions on a compact space, the space
of $L^{\infty}$ functions on a $\sigma$-finite measure space, and the space of
bounded linear operators on a Hilbert space. In this paper, we focus on the
$C^{*}$-algebra of the space of continuous functions on a compact space.
Practically, we can use tools for functions such as regression and integration
to learn multiple models efficiently and to adapt the models to problems
continuously. Theoretically, our new framework highlights the potential
possibility of applying the theory of $C^{*}$-algebra to general neural
network models. Fig. 1 shows an overview of our framework schematically. The
application of $C^{*}$-algebra to data analysis is discussed by Hashimoto et
al. (2021). We focus on the practical applications to neural network in this
paper. To the best of our knowledge, this is the first paper that applies the
theory of $C^{*}$-algebra to neural network models.

Our contributions are as follows:
$\bullet$ We propose a generic framework of neural network with
$C^{*}$-algebra-valued parameters.
$\bullet$ We propose a gradient descent method to learn the model on
$C^{*}$-algebra.
$\bullet$ We apply our framework to practical problems such as density
estimation and few-shot learning.
Regarding the second one, since $C^{*}$-algebra admits a generalization of
Hilbert space, which is called Hilbert $C^{*}$-module, we can generalize the
classical gradient descent method on the Hilbert space associated with the
parameters of a model to that on the Hilbert $C^{*}$-module associated with
the $C^{*}$-algebra-valued parameters.

The remainder of this paper is organized as follows. In Section 3, we review
mathematical notions related to $C^{*}$-algebra and Hilbert $C^{*}$-module. In
Section 4, we propose a model with $C^{*}$-algebra-valued parameters. Then, in
Section 5, we discuss practical applications and show numerical results. We
conclude the paper in Section 6. The source code of this paper is available at
https://www.rd.ntt/e/ns/qos/person/hashimoto/code_c_star_net.zip.

##### Notations

Bold letters denote $\mathbb{R}$-valued objects or maps from
$\mathbb{R}^{d_{1}}$ to $\mathbb{R}^{d_{2}}$ for some
$d_{1},d_{2}\in\mathbb{N}$. Italic letters denote $\mathcal{A}$-valued objects
or maps from $\mathcal{A}^{d_{1}}$ to $\mathcal{A}^{d_{2}}$.

Figure 1: Overview of our framework.

##  2 Motivation of applying $C^{*}$-algebra to neural networks

Because $C^{*}$-algebra and Hilbert $C^{*}$-module are natural generalizations
of the space of complex numbers and Hilbert space, we can naturally generalize
neural networks by using $C^{*}$-algebra. Let $\mathcal{A}$ be the
$C^{*}$-algebra of the space of continuous functions on a compact space. Then,
by generalizing the real-valued parameters in the neural networks to
$C^{*}$-algebra-valued ones, we can combine multiple real-valued models
continuously. For example, we can apply our framework to ensemble learning. In
standard ensemble learning, we learn multiple models separately and aggregate
the results of the models after finishing the learning process. This approach
may not be efficient since the multiple models do not interact during the
learning process, although each model is learned for the same or related task.
On the other hand, by using the $\mathcal{A}$-valued parameters, we introduce
a continuous dependence between different models and can learn the multiple
models with interactions. As a result, we expect that our method outperforms
the standard ensemble learning. For further details of applications of our
framework to practical situations, see Section 5.

##  3 Background

In this section, we review mathematical notions required for the remaining
part of this paper. In Subsection 3.1, we review $C^{*}$-algebra, and in
Subsection 3.2, we review Hilbert $C^{*}$-module. All the definitions and
examples in this section are standard terminologies in $C^{*}$-algebra, and
they are adopted from Hashimoto et al. (2021). The standard definitions
related to the definitions in this section is provided in Section A. For
further details of $C^{*}$-algebra and Hilbert $C^{*}$-module, see (Lance,
1995; Murphy, 1990; Hashimoto et al., 2021).

###  3.1 $C^{*}$-algebra

$C^{*}$-algebra is a generalization of the space of complex numbers.

######  Definition 3.1 ($C^{*}$-algebra).

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

  1. 1.

$\mathcal{A}$ is an algebra (See Definition A.1) 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 $\|\cdot\|$, and for $c,d\in\mathcal{A}$,
$\|cd\|\leq\|c\|\,\|d\|$ holds. In addition, $\mathcal{A}$ is complete with
respect to $\|\cdot\|$.

  3. 3.

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

We introduce important notions related to $C^{*}$-algebra.

######  Definition 3.2 (Multiplicative identity).

The multiplicative identity of a $C^{*}$-algebra $\mathcal{A}$ is the element
$a\in\mathcal{A}$ that satisfies $ac=ca=c$ for any $c\in\mathcal{A}$. We
denote by $1_{\mathcal{A}}$ the multiplicative identity of $\mathcal{A}$.

######  Definition 3.3 (Positive).

An element $c$ of $\mathcal{A}$ is called positive if there exists
$d\in\mathcal{A}$ such that $c=d^{*}d$ holds. We denote by $\mathcal{A}_{+}$
the subset of $\mathcal{A}$ composed of all positive elements in
$\mathcal{A}$.

An important example of $C^{*}$-algebras is the space of continuous functions
on a compact space, on which we focus in this paper.

######  Example 3.4.

Let $\mathcal{Z}$ be a compact space and let $C(\mathcal{Z})$ be the Banach
space of continuous functions equipped with the sup norm. Let
$\cdot:C(\mathcal{Z})\times C(\mathcal{Z})\to C(\mathcal{Z})$ be defined as
$(c\cdot d)(z)=c(z)d(z)$ for $c,d\in C(\mathcal{Z})$ and $z\in\mathcal{Z}$. In
addition, let $(\cdot)^{*}:C(\mathcal{Z})\to C(\mathcal{Z})$ be defined as
$c^{*}(z)=\overline{c(z)}$ for $c\in C(\mathcal{Z})$. Then, $C(\mathcal{Z})$
is a $C^{*}$-algebra. The multiplicative identity is the constant function
whose value is $1$ at any $z\in\mathcal{Z}$. For $c\in C(\mathcal{Z})$, $c$ is
positive if and only if $c(z)\geq 0$ for any $z\in\mathcal{Z}$.

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

Hilbert $C^{*}$-module is a generalization of Hilbert space. We first
introduce $C^{*}$-module, which is a generalization of vector space. Then, we
introduce $\mathcal{A}$-valued inner product and Hilbert $C^{*}$-module.

######  Definition 3.5 ($C^{*}$-module).

Let $\mathcal{M}$ be an abelian group with operation $+$ and let $\mathcal{A}$
be a $C^{*}$-algebra. If $\mathcal{M}$ is equipped with an
$\mathcal{A}$-multiplication (See Definition A.2), $\mathcal{M}$ is called a
$C^{*}$-module over $\mathcal{A}$.

######  Definition 3.6 ($\mathcal{A}$-valued inner product).

Let $\mathcal{A}$ be a $C^{*}$-algebra and let $\mathcal{M}$ be a
$C^{*}$-module over $\mathcal{A}$. A $\mathbb{C}$-linear map with respect to
the second variable
$\left\langle\cdot,\cdot\right\rangle:\mathcal{M}\times\mathcal{M}\to\mathcal{A}$
is called an $\mathcal{A}$-valued inner product if it satisfies the following
conditions for $u,v,p\in\mathcal{M}$ and $c,d\in\mathcal{A}$:

$\bullet$ $\left\langle u,vc+pd\right\rangle=\left\langle u,v\right\rangle
c+\left\langle u,p\right\rangle d$, $\bullet$ $\left\langle
v,u\right\rangle=\left\langle u,v\right\rangle^{*}$,
$\bullet$ $\left\langle u,u\right\rangle$ is positive, $\bullet$ If
$\left\langle u,u\right\rangle=0$ then $u=0$.

######  Definition 3.7 (Norm).

Let $\mathcal{A}$ be a $C^{*}$-algebra and let $\mathcal{M}$ be a
$C^{*}$-module over $\mathcal{A}$ equipped with an $\mathcal{A}$-valued inner
product $\left\langle\cdot,\cdot\right\rangle$. The (real-valued) norm
$\|\cdot\|$ on $\mathcal{M}$ is defined by $\|u\|=\big{\|}\left\langle
u,u\right\rangle\big{\|}_{\mathcal{A}}^{1/2}$, where $\|\cdot\|_{\mathcal{A}}$
is the norm in $\mathcal{A}$.

######  Definition 3.8 (Hilbert $C^{*}$-module).

Let $\mathcal{A}$ be a $C^{*}$-module. Let $\mathcal{M}$ be a $C^{*}$-module
over $\mathcal{A}$ equipped with an $\mathcal{A}$-valued inner product defined
in Definition 3.6. If $\mathcal{M}$ is complete with respect to the norm
$\|\cdot\|$ defined in Definition 3.7, it is called a Hilbert $C^{*}$-module
over $\mathcal{A}$ or Hilbert $\mathcal{A}$-module.

##  4 Neural network on $C^{*}$-algebra

In this section, we propose a new framework with $C^{*}$-algebra that
generalizes the existing framework of neural networks. In Subsection 4.1, we
formulate the existing framework of neural networks. Then, in Subsection 4.2,
we generalize the existing framework to that on $C^{*}$-algebra.

###  4.1 Existing framework of neural networks

We begin by formulating the existing framework of neural networks. We consider
a network with $H\in\mathbb{N}$ hidden layers. Let $N_{0},\ldots,{N_{H+1}}$ be
natural numbers each of which represents the dimension of a layer. Inputs are
vectors in $\mathbb{R}^{N_{0}}$ and outputs are vectors in
$\mathbb{R}^{N_{H+1}}$. In addition, for $i=1,\ldots,H+1$, let
$\mathbf{W}_{i}:\mathbb{R}^{N_{i-1}}\to\mathbb{R}^{N_{i}}$ be an
$N_{i-1}\times N_{i}$ matrix and
$\boldsymbol{\sigma}_{i}:\mathbb{R}^{N_{i}}\to\mathbb{R}^{N_{i}}$ be an (often
nonlinear) activation function. The neural network model
$\mathbf{f}:\mathbb{R}^{N_{0}}\to\mathbb{R}^{N_{H+1}}$ is defined as

$\mathbf{f}=\boldsymbol{\sigma}_{H+1}\circ\mathbf{W}_{H+1}\circ\boldsymbol{\sigma}_{H}\circ\mathbf{W}_{H}\circ\cdots\circ\boldsymbol{\sigma}_{1}\circ\mathbf{W}_{1}.$
(1)

We fix $\boldsymbol{\sigma}_{1},\ldots,\boldsymbol{\sigma}_{H+1}$ and find
best possible matrices $\mathbf{W}_{1},\ldots,\mathbf{W}_{H+1}$ by minimizing
a loss function $\mathbf{L}$. We regard the set of matrices
$\mathbf{W}_{1},\ldots,\mathbf{W}_{H+1}$ as an $N$-dimensional vector of
parameters, which is denoted as $\boldsymbol{\theta}$, where
$N=\sum_{i=1}^{H+1}N_{i-1}N_{i}$. Then, we set a loss function
$\mathbf{L}:\mathbb{R}^{N}\to\mathbb{R}_{+}$, which depends on the parameters
(and usually on inputs and outputs). Here, $\mathbb{R}_{+}$ is the set of all
non-negative real numbers. We learn the parameter $\boldsymbol{\theta}$ by
minimizing the loss function. The minimization of the loss function is
implemented by a gradient descent method such as stochastic gradient descent
(SGD) and Adam (Kingma & Ba, 2015). We compute the gradient
$\nabla_{\boldsymbol{\theta}}\mathbf{L}$ of the loss function $\mathbf{L}$ and
generate a sequence $\boldsymbol{\theta}_{0},\boldsymbol{\theta}_{1},\ldots$
using the gradient for finding a best possible $\boldsymbol{\theta}$.

###  4.2 Generalization to $C^{*}$-algebra

####  4.2.1 Formulation

To improve the representational power of the model, we generalize the
parameter $\theta$ on $\mathbb{R}^{N}$, which is a Hilbert space, to a Hilbert
$C^{*}$-module. Let $\mathcal{A}$ be a commutative unital $C^{*}$-algebra. By
the Gelfand–Naimark theorem, there exists a compact Hausdorff space
$\mathcal{Z}$ such that $\mathcal{A}$ is isometrically $*$-isomorphic to the
$C^{*}$-algebra $C(\mathcal{Z})$, the space of continuous functions on
$\mathcal{Z}$ (see Example 3.4). Therefore, we focus on the case of
$\mathcal{A}=C(\mathcal{Z})$ for a compact Hausdorff space $\mathcal{Z}$ in
the remaining parts of this paper. As before, let $N_{0},\ldots,{N_{H+1}}$ be
natural numbers each of which represents the dimension of a layer. However, in
this case, we consider $\mathcal{A}^{N_{0}}$-valued inputs and
$\mathcal{A}^{N_{H+1}}$-valued outputs. Since $\mathcal{A}$ is a function
space, the inputs and outputs should be functions, which enables us to analyze
functional data. On the other hand, for scalar-valued data
$\mathbf{x}\in\mathbb{R}$, we can transform $\mathbf{x}$ into an appropriate
function such as the constant function ${x}\equiv\mathbf{x}$ (see Subsections
5.1 and 5.2 for practical applications). In addition, for $i=1,\ldots,H+1$,
let $W_{i}:\mathcal{A}^{N_{i-1}}\to\mathcal{A}^{N_{i}}$ be an $N_{i-1}\times
N_{i}$ $\mathcal{A}$-valued matrix and
$\sigma_{i}:\mathcal{A}^{N_{i}}\to\mathcal{A}^{N_{i}}$ be an (often nonlinear)
activation function. The neural network model
$f:\mathcal{A}^{N_{0}}\to\mathcal{A}^{N_{0}}$ is defined as

$f=\sigma_{H+1}\circ W_{H+1}\circ\sigma_{H}\circ
W_{H}\circ\cdots\circ\sigma_{1}\circ W_{1}$ (2)

in the same manner as Eq. (1). We regard the set of $\mathcal{A}$-valued
matrices $W_{1},\ldots,W_{H+1}$ as an $N$-dimensional $\mathcal{A}$-valued
vector of parameters, which is denoted as $\theta$, where
$N=\sum_{i=1}^{H+1}N_{i-1}N_{i}$. Then, we set an $\mathcal{A}$-valued loss
function $L:\mathcal{A}^{N}\to\mathcal{A}_{+}$, which depends on the
$\mathcal{A}$-valued parameters. Here, $\mathcal{A}_{+}$ is the set of all
positive elements in $\mathcal{A}$ (see Definition 3.3).

####  4.2.2 Learning $C^{*}$-algebra-valued parameters

To implement a gradient descent method on $\mathcal{A}^{N}$ and minimize the
$\mathcal{A}$-valued loss function, we propose a practical approach to
applying the gradient descent method proposed by Hashimoto et al. (2021) to
our case. In more detail, we add the regularization term and simultaneously
learn multiple models with interactions. We first define a gradient of
$\mathcal{A}$-valued functions on $\mathcal{A}^{N}$.

######  Definition 4.1 ($\mathcal{A}$-valued gradient).

Let $L:\mathcal{A}^{N}\to\mathcal{A}$ be an $\mathcal{A}$-valued function
defined on $\mathcal{A}^{N}$ and let $\theta\in\mathcal{A}^{N}$. Assume there
exists $\xi\in\mathcal{A}^{N}$ such that for any $\delta\in\mathcal{A}^{N}$
and any $z\in\mathcal{Z}$,

$\lim_{\delta\to 0,\
\delta\in\mathcal{A}^{N}\setminus\\{0\\}}\frac{L(\theta+\delta)(z)-L(\theta)(z)-\left\langle\xi,\delta\right\rangle(z)}{\|\delta\|}=0.$

In this case, we define $\xi$ as the $\mathcal{A}$-valued gradient of $L$ at
$\theta$ and denote it by $\nabla_{\mathcal{A},\theta}L$.

######  Example 4.2.

Assume there exists a function
$\tilde{L}:\mathbb{R}^{N}\times\mathcal{Z}\to\mathbb{R}$ such that
$L(\theta)(z)=\tilde{L}(\theta(z),z)$, that is, we can decompose $L$ into
$\mathbb{R}$-valued functions indexed by $z$ on $\mathbb{R}^{N}$. The function
$\tilde{L}(\cdot,z)$ corresponds to an $\mathbb{R}$-valued (standard) loss
function at $z\in\mathcal{Z}$. Assume $\tilde{L}(\cdot,z)$ has the (standard)
gradient $\nabla_{\theta(z)}\tilde{L}(\cdot,z)\in\mathbb{R}^{N}$ for each
$z\in\mathcal{Z}$. If the map $z\mapsto\nabla_{\theta(z)}\tilde{L}(\cdot,z)$
is contained in $\mathcal{A}^{N}$, then it is the $\mathcal{A}$-valued
gradient.

We generate a sequence $\theta_{0},\theta_{1}\ldots$ using the
$\mathcal{A}$-valued gradient for finding a best possible $\theta$ to minimize
the $\mathcal{A}$-valued loss function $L$. The basic gradient descent scheme
is

$\displaystyle\theta_{0}\in\mathcal{A}^{N},$ (3)
$\displaystyle\theta_{t+1}=\theta_{t}-\nabla_{\mathcal{A},\theta_{t}}L\cdot\eta_{t}\
(t=0,1,\ldots),$

where $\eta_{t}\in\mathcal{A}_{+}$ is the learning rate.

######  Example 4.3.

Assume there exists a function
$\tilde{L}:\mathbb{R}^{N}\times\mathcal{Z}\to\mathbb{R}$ such that
$L(\theta)(z)=\tilde{L}(\theta(z),z)$. Then, for each $z\in\mathcal{Z}$, the
scheme (3) is reduced to

$\displaystyle\theta_{0}(z)\in\mathbb{R}^{N},$
$\displaystyle\theta_{t+1}(z)=\theta_{t}(z)-\eta_{t}(z)\nabla_{\theta_{t}(z)}\tilde{L}(\cdot,z)\
(t=0,1,\ldots),$

which is the standard gradient descent scheme on $\mathbb{R}^{N}$ with the
learning rate $\eta_{t}(z)$. Thus, computing the scheme (3) is equivalent to
computing the standard gradient descent scheme on $\mathbb{R}^{N}$
simultaneously for all $z\in\mathcal{Z}$ in this case. If the standard
gradient descent at each $z\in\mathcal{Z}$ generates a sequence
$\theta_{0}(z),\theta_{1}(z),\ldots$ in $\mathbb{R}^{N}$ converging to some
$\theta^{*}(z)$, then we obtain a sequence $\theta_{0},\theta_{1},\ldots$ in
$\mathcal{A}^{N}$ converging pointwise to the function $\theta^{*}$.

The above example implies that if $L(\theta)$ is defined as
$L(\theta)(z)=\tilde{L}(\theta(z),z)$ for a function
$\tilde{L}:\mathbb{R}^{N}\times\mathcal{Z}\to\mathbb{R}$, then the scheme (3)
is computed without any interactions among variables $z\in\mathcal{Z}$. To
learn the model with interactions among $z$, we assume $\mathcal{Z}$ is a
compact finite measure space, and add an $L_{1}$ regularization term to the
loss function $L$ as

$L{\operatorname{reg}}(\theta)=L(\theta)+\int_{z\in\mathcal{Z}}L(\theta)(z)dz1_{\mathcal{A}}\cdot\lambda,$

where $1_{\mathcal{A}}$ is the multiplicative identity of $\mathcal{A}$
($1_{\mathcal{A}}\equiv 1$, see Definition 3.2) and
$\lambda\in\mathcal{A}_{+}$ is a hyperparameter. Since the regularization term
is a constant function with respect to $z$, it affects uniformly on
$L_{\operatorname{reg}}(\theta)$ at any $z\in\mathcal{Z}$. It has an effect of
aggregating the gradient at each $z\in\mathcal{Z}$ and adding the aggregated
gradient to the gradient at each $z\in\mathcal{Z}$. Indeed, since
$\nabla_{\mathcal{A},\theta}L\in\mathcal{A}^{N}$, each element of
$\nabla_{\mathcal{A},\theta}L$ is integrable. Thus, we have

$\nabla_{\mathcal{A},\theta}L_{\operatorname{reg}}(\theta)=\nabla_{\mathcal{A},\theta}L+\int_{z\in\mathcal{Z}}(\nabla_{\mathcal{A},\theta}L)(z)dz\odot\mathbf{1}_{\mathcal{A}}\lambda,$

where
$\mathbf{1}_{\mathcal{A}}=[1_{\mathcal{A}},\ldots,1_{\mathcal{A}}]^{T}\in\mathcal{A}^{N}$
and $\odot$ represents the element-wise product of
$\int_{z\in\mathcal{Z}}(\nabla_{\mathcal{A},\theta})L(z)dz\in\mathbb{C}^{N}$
and $\mathbf{1}_{\mathcal{A}}\in\mathcal{A}^{N}$.

In practical computations, we cannot handle functions in the infinite-
dimensional space $\mathcal{A}$. Therefore, we set a finite-dimensional
subspace $\mathcal{V}$ of $\mathcal{A}^{N}$ and a map
$P:\mathcal{A}^{N}\to\mathcal{V}$. Moreover, we set the learning rate
$\eta_{t}$ and the hyperparameter $\lambda$ as constant functions
$\eta_{t}\equiv\tilde{\eta}_{t}$ and $\lambda\equiv\tilde{\lambda}$ for some
$\tilde{\eta}_{t},\tilde{\lambda}\in\mathbb{R}+$. Then, the practical scheme
is

$\displaystyle\theta_{0}\in\mathcal{V},$ (4)
$\displaystyle\theta_{t+1}=\theta_{t}-\tilde{\eta}_{t}P(\nabla_{\mathcal{A},\theta_{t}}L)$
$\displaystyle\quad-\tilde{\lambda}\int_{z\in\mathcal{Z}}P(\nabla_{\mathcal{A},\theta}L)(z)dz\odot\mathbf{1}_{\mathcal{A}}\
(t=0,1,\ldots).$

Typically, we set $P(\theta)$ as a regression of
$\theta(z_{1}),\ldots,\theta(z_{n})$ in $\mathcal{V}$, where
$z_{1},\ldots,z_{n}$ for some $n\in\mathbb{N}$ are properly chosen fixed
points. As a result, the sequence $\theta_{0},\theta_{1},\ldots$ is contained
in $\mathcal{V}$, which enables us to compute the scheme (4) practically. In
addition, by applying $P$, the scheme becomes interactive with respect to
$z\in\mathcal{Z}$. That is, $P(\theta)(z_{0})$ is determined by using the
values $\theta(z)$ for $z\neq z_{0}$. Regarding the choice of $\mathcal{V}$,
we choose it on the basis of the following fact.

######  Fact 4.4.

Let $\mathcal{Z}$ be a compact metric space and let $d_{\mathcal{Z}}$ be the
metric on $\mathcal{Z}$. Let $\zeta_{0},\zeta_{1},\ldots\in\mathcal{A}$ be a
sequence such that there exists a function $\zeta^{*}$ on $\mathcal{Z}$,
$\lim_{t\to\infty}\zeta_{t}(z)=\zeta^{*}(z)$ for any $z\in\mathcal{Z}$. If
$\zeta_{0},\zeta_{1},\ldots$ is uniformly Lipschitz continuous, that is, there
exists $C>0$ such that for any $t\in\mathbb{N}$,

$|\zeta_{t}(z_{1})-\zeta_{t}(z_{2})|\leq Cd_{\mathcal{Z}}(z_{1},z_{2}),$

then $\zeta_{0},\zeta_{1},\ldots$ converges uniformly to $\zeta^{*}$ and
$\zeta^{*}\in\mathcal{A}$.

On the basis of Fact 4.4, we set
$\mathcal{V}=\operatorname{Span}\\{v_{1},\ldots,v_{l}\\}^{N}$ for
$l\in\mathbb{N}$ and Lipschitz continuous functions
$v_{1},\ldots,v_{l}\in\mathcal{A}$. Then, each element
$\theta_{t,j}\in\mathcal{A}$ of $\theta_{t}$ is represented as
$\sum_{i=1}^{n}c_{t,i}v_{i}$ for some $c_{t,i}\in\mathbb{C}$. If there exists
$C_{i}>0$ such that for any $t\in\mathbb{N}$ $|c_{t,i}|\leq C_{i}$ for
$i=1,\ldots,n$, then the sequence $\theta_{0,j},\theta_{1,j},\ldots$ is
uniformly Lipschitz. Thus, if it converges pointwise, the convergence is
uniform. Uniform convergence is a stronger notion than pointwise convergence
and preserves properties of the sequence of functions to its limit. For
example, the uniform limit of a sequence of uniformly continuous functions is
also uniformly continuous, but the pointwise limit of a sequence of continuous
functions is not always continuous. Thus, the uniform convergence is more
suitable when we need the parameter $\theta$ to preserve properties as a
function throughout the gradient descent.

######  Remark 4.5.

The scheme 4 is a basic scheme of the $\mathcal{A}$-valued gradient descent.
We can also consider variants of the scheme such as SGD and Adam in the same
manner as the standard gradient descent on $\mathbb{R}^{N}$.

##### Computation complexity

The computation complexity of learning the $C^{*}$-algebra net is $O(pl)$,
where $p$ is the number of parameters of the network and $l$ is the number of
basis functions in $\mathcal{V}$. It is $l$ times as large as that of the
corresponding $\mathbb{R}$-valued model.

We illustrate the difference between $\mathbb{R}$-valued and
$\mathcal{A}$-valued models for a specific case of the linear regression
problem on $\mathbb{R}$ in Section C.

##  5 Applications

In this section, we show examples of practical applications of our framework.
In Subsection 5.1, we apply our framework to density estimation with
normalizing flow. In Subsection 5.2, we apply our framework to few-shot
learning. By using a model with function-valued parameters, we can learn
features of data efficiently even with a limited number of samples. Our
framework is general and its application is not restricted to the above two
cases. In Subsection 5.3, we show other examples of applications.

###  5.1 Density estimation

####  5.1.1 Density estimation with normalizing flow

We can improve density estimation by replacing parameters in a model by
$\mathcal{A}$-valued ones. In this paper, we focus on using normalizing flow
(Dinh et al., 2014, 2017; Teshima et al., 2020), but the application is not
limited to normalizing flow. We first review density estimation with
normalization flow briefly. Let $\Omega$ be a probability space. We construct
$\mathbf{f}=\mathbf{f}^{\boldsymbol{\theta}}$ in Eq. (1) as a measurable,
invertible, and differentiable map that transforms a normal distribution into
the distribution of data. That is, if the distribution of a random variable
$X$ defined on $\Omega$ and taking its value in $\mathbb{R}^{N_{0}}$ is the
normal distribution, then the distribution of
$\mathbf{f}^{\boldsymbol{\theta}}(X)$ is the distribution of data. The density
$p^{\boldsymbol{\theta}}_{\operatorname{data}}$ of
$\mathbf{f}^{\boldsymbol{\theta}}(X)$ is calculated as

$p^{\boldsymbol{\theta}}_{\operatorname{data}}(\mathbf{x})=p_{\operatorname{n}}\big{(}(\mathbf{f}^{\boldsymbol{\theta}})^{-1}(\mathbf{x})\big{)}\big{|}\;\operatorname{det}\big{(}\nabla_{\mathbf{x}}(\mathbf{f}^{\boldsymbol{\theta}})^{-1}\big{)}\big{|}$
(5)

for $x\in\mathbb{R}^{N_{0}}$, where $p_{\operatorname{n}}$ is the density of
the normal distribution. By using the formula (5), we compute the likelihood
of given samples $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{N_{0}}$
and set the loss function $\mathbf{L}:\mathbb{R}^{N}\to\mathbb{R}_{+}$ as a
negative log likelihood:

$\mathbf{L}(\boldsymbol{\theta})=-\sum_{i=1}^{n}\log
p^{\boldsymbol{\theta}}_{\operatorname{data}}(\mathbf{x}_{i}).$

####  5.1.2 Generalization to $C^{*}$-algebra

We generalize the above setting to that with $C^{*}$-algebra. Let
$\mathcal{Z}$ be a compact probability space and let $X$ be a random variable
defined on $\mathcal{Z}\times\Omega$ and taking its value in
$\mathbb{R}^{N_{0}}$. Assume that for any $\omega\in\Omega$, $X(\cdot,\omega)$
is continuous on $\mathcal{Z}$. Let $\mathcal{D}$ be the probability measure
on $\mathcal{Z}$. The map $f=f^{\theta}$ in Eq. (2) is constructed so that if
for any $z\in\mathcal{Z}$, the distribution of a random variable $X(z,\cdot)$
is a normal distribution on $\mathbb{R}^{N_{0}}$, then the distribution of the
random variable $\omega\mapsto f^{\theta}(X(\cdot,\omega))(z)$ is the
distribution of data. The density $p^{\theta,z}_{\operatorname{data}}$ of the
random variable $\omega\mapsto f^{\theta}(X(\cdot,\omega))(z)$ is calculated
as

$p^{\theta,z}_{\operatorname{data}}(\mathbf{x})=p_{\operatorname{n}}^{z}\big{(}(f^{\theta})^{-1}({x})(z)\big{)}\;\big{|}\operatorname{det}\big{(}\nabla_{\mathcal{A},{x}}(f^{\theta})^{-1}(z)\big{)}\big{|}$
(6)

for $\mathbf{x}\in\mathbb{R}^{N_{0}}$, where ${x}\in\mathcal{A}^{N_{0}}$ is
the constant function ${x}\equiv\mathbf{x}$. By using the formula (6), we
compute the likelihood of given samples
$\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{N_{0}}$ and set the loss
function $L:\mathcal{A}^{N}\to\mathcal{A}_{+}$ as a negative log likelihood:

$\displaystyle L(\theta)(z)=-\sum_{i=1}^{n}\log
p^{\theta,z}_{\operatorname{data}}(\mathbf{x}_{i}).$ (7)

We apply the $\mathcal{A}$-valued gradient descent method proposed in
Subsection 4.2.2 to minimize the loss function (7). Since $L(\theta)(z)$
depends only on $\theta(z)$, we have $L(\theta)(z)=\tilde{L}(\theta(z),z)$ for
some function $\tilde{L}:\mathbb{R}^{N}\times\mathcal{Z}\to\mathbb{R}$. As we
explained in Example 4.3, the $\mathcal{A}$-valued gradient of $L$ is
calculated by the standard gradient of $\tilde{L}(\cdot,z)$. After learning
the model, we obtain the distribution ${p}^{\theta,z}_{\operatorname{data}}$
at each $z\in\mathcal{Z}$. Ideally, ${p}^{\theta,z}_{\operatorname{data}}$ is
independent of $z$ since the distribution of data is independent of $z$.
Practically, we get an estimation of the density
$\tilde{p}^{\theta}_{\operatorname{data}}$ of data by integrating
$p^{\theta,z}_{\operatorname{data}}$ as
$\tilde{p}^{\theta}_{\operatorname{data}}(\mathbf{x})=\int_{z\in\mathcal{Z}}p^{\theta,z}_{\operatorname{data}}(\mathbf{x})d\mathcal{D}(z)$.

(a) Existing

(b) Ours

Figure 2: Difference between the existing method and our method of density
estimation with normalizing flow. (Although the set $\mathcal{Z}$ in (b) is
depicted as if it is a one-dimensional space, we can set $\mathcal{Z}$ as an
arbitrary compact space.)

(a) Swiss roll

(b) Circles

(c) Swiss roll (training)

(d) Circles (training)

Figure 3: Samples in the datasets ((c) and (d) show training samples)

####  5.1.3 Numerical results

|  |  |
---|---|---|---
|  |  |
Standard |  |  |
Density |  |  |
|  |  |
NLL |  $3.68\pm 0.413$ |  $4.07\pm 0.340$ |  $3.91\pm 0.621$
|  |  |
|  |  |
Discrete |  |  |
Density |  |  |
|  |  |
NLL |  $4.09\pm 0.341$ |  $4.00\pm 0.198$ |  $4.03\pm 0.268$
|  |  |
|  |  |
Ours |  |  |
Density |  |  |
|  |  |
NLL |  $3.75\pm 0.022$ |  $\mathbf{2.97\pm 0.029}$ |  $\mathbf{2.94\pm 0.057}$
|  $2000$ epochs |  $2500$ epochs |  $3000$ epochs
(a) Swiss roll

|  |  |
---|---|---|---
|  |  |
Standard |  |  |
Density |  |  |
|  |  |
NLL |  $4.81\pm 0.211$ |  $4.73\pm 0.556$ |  $4.62\pm 0.254$
|  |  |
|  |  |
Discrete |  |  |
Density |  |  |
|  |  |
NLL |  $4.91\pm 0.129$ |  $5.00\pm 0.135$ |  $5.14\pm 0.200$
|  |  |
|  |  |
Ours |  |  |
Density |  |  |
|  |  |
NLL |  $4.97\pm 0.210$ |  $\mathbf{4.17\pm 0.245}$ |  $\mathbf{4.17\pm 0.157}$
|  $2000$ epochs |  $2500$ epochs |  $3000$ epochs
(b) Circles

Figure 4: Comparison among estimated densities and negative log-likelihoods
(NLLs) with existing methods and our method. (Each value of negative log
likelihood is an average ($\pm$ a standard deviation) over 5 independent
runs.)

We show the validity of applying our method to density estimation with
normalizing flow numerically. We generated two toy datasets: swiss roll and
circles. Each dataset contains $1000$ samples, illustrated in Fig. 3. We put
$100$ samples for training samples from the datasets. The estimation of the
densities is challenging since the amount of training samples is small. We
used masked autoregressive flow (Papamakarios et al., 2017) to construct the
invertible differentiable map $f^{\theta}$, and for the gradient descent
method, we used Adam. We set the learning rate so that it decays polynomially
starting from $0.001$ with decay rate $0.5$. We set
$\mathcal{Z}=[-4,4]\times[-4,4]\subseteq\mathbb{R}^{2}$ and set the finite-
dimensional subspace $\mathcal{V}$ of $\mathcal{A}^{N}$ as
$\operatorname{Span}\\{v_{1},\ldots,v_{l}\\}^{N}$, where $l=9$,
$v_{i}(z)=e^{-10\|z_{i}-z\|^{2}}$, $z_{1}=[0,0]$, and
$z_{4i+j+1}=[(2+i)\sin(2\pi(j-1+0.5i)/4),(2+i)\cos(2\pi(j-1+0.5i)/4)]$ for
$i=0,1$ and $j=1,\ldots,4$. The density $p_{\operatorname{n}}(z,\cdot)$ of the
base normal distribution at $z\in\mathcal{Z}$ is set as the normal
distribution with mean $z$ and standard deviation $1$. We set the map
${P}:\mathcal{A}\to\mathcal{V}$ as the kernel ridge regression, where the
$i$th element of ${P}(\theta)$ for
${\theta}=[\theta_{1},\ldots,\theta_{N}]^{T}\in\mathcal{A}^{N}$ is computed as
$[v_{1},\ldots,v_{n}](G+\mu
I)^{-1}[{\theta}_{i}(z_{1}),\ldots,{\theta}_{i}(z_{l})]^{T}$. Here, $G$ is the
Gram matrix whose $(i,j)$-element is $e^{-10\|z_{i}-z_{j}\|^{2}}$ and
$\mu\in\mathbb{R}_{+}$ is a hyperparameter of the regression that controls the
strength of a regularization term of the regression. In this experiment, we
set $\mu=0.1$. In addition, we set the hyperparameter $\tilde{\lambda}$ in Eq.
(4) as $0.3$ and set the distribution $\mathcal{D}$ on $\mathcal{Z}$ as the
uniform distribution on $\bigcup_{i=1}^{9}\\{z\in\mathcal{Z}\ \mid\
\|z-z_{i}\|\leq 0.05\\}$. We compared our method proposed in Subsection 5.1.2
with two straightforward methods, (1) the standard method explained in
Subsection 5.1.1, called “standard”, and (2) learning the model whose base
normal distribution is $p_{\operatorname{n}}(z_{i},\cdot)$ for $i=1,\ldots 9$
separately and computing the mean value of the results, called “discrete”.
Note that the method (1) corresponds to the case of $l=1$ and
$\tilde{\lambda}=\mu=0$ and the method (2) corresponds to the case of $l=9$
and $\tilde{\lambda}=\mu=0$ of the proposed method.

Fig. 4 shows the estimated densities and negative log-likelihoods computed
with the $900$ samples other than the training samples in each dataset. We see
that our method with 2500 or 3000 epochs gives the best estimation of the
densities, and its negative log-likelihoods are the smallest for each dataset.
Since the amount of samples is small, the standard method fails to capture the
detailed shape of the density and its shape is blurred. By learning multiple
models separately and computing the mean value, the shape becomes more clear,
but not smooth. Our method captures the shape clearly and smoothly because it
allows us to learn multiple models simultaneously with interactions by using
smooth functions.

##### Computation complexity and memory usage

The method “discrete” learns $9$ real-valued models separately, and in the
above experiment, we compared our proposed method with $l=9$ to the method
“discrete”. The computation complexities of these two methods are the same.
Precisely, in our method, we compute the approximation of elements in
$\mathcal{A}$ when updating the parameters. (The operator $P$ in Eq. (4))
However, the computation complexity of this part can be ignored since we used
the kernel ridge regression and $l\ll p$. The approximation of a function
$a\in\mathcal{A}$ is computed by just the multiplication of an $l\times l$
fixed matrix to the vector $[a(z_{1}),\ldots,a(z_{l})]$. Our method
outperforms the method “Discrete” since our method properly combines multiple
models continuously. Regarding the memory usage, the amount of memory only
increases linearly with respect to $l$.

###  5.2 Few-shot learning

####  5.2.1 Few-shot learning for classification

We can improve the accuracy of few-shot learning by replacing parameters in a
model by $\mathcal{A}$-valued ones. In this paper, we focus on classification
tasks, but the application of our framework is not limited to classification
tasks. Few-shot learning challenges a model to be learned with a limited
number of samples (Fei-Fei et al., 2006; Lake et al., 2011). We first
formulate the setting for a supervised classification task briefly. We
construct $\mathbf{f}=\mathbf{f}^{\boldsymbol{\theta}}$ in Eq. (1) as a map
which maps a sample such as an image to its label, that is, the probability to
belong to each class. For given samples
$\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{N_{0}}$ and their labels
$\mathbf{y}_{1},\ldots,\mathbf{y}_{n}\in\mathbb{R}^{N_{H+1}}$, we set the loss
function $\mathbf{L}:\mathbb{R}^{N}\to\mathbb{R}_{+}$ as the cross categorical
entropy:

$\mathbf{L}(\boldsymbol{\theta})=-\sum_{i=1}^{n}\left\langle\mathbf{y}_{i},\log(\mathbf{f}^{\boldsymbol{\theta}}(\mathbf{x}_{i}))\right\rangle.$

For few-shot learning, one approach to learning the model effectively is meta-
learning (Ravi & Larochelle, 2017; Finn et al., 2017; Rusu et al., 2019).
Before learning a model for a new task $\mathcal{T}_{\operatorname{new}}$, we
learn a meta-model with multiple tasks
$\mathcal{T}_{1},\ldots,\mathcal{T}_{m}$ to extract common features among all
tasks. For example, Rusu et al. (2019) propose to learn a model to obtain a
map $Z$ that maps a task $\mathcal{T}_{i}$ to its low-dimensional
representation $z_{i}$ and a map $\Theta$ that maps $z_{i}$ into the parameter
of a model specific to the task $\mathcal{T}_{i}$. Then, for a given new task
$\mathcal{T}_{\operatorname{new}}$, we can get a corresponding low-dimensional
representation $z_{\operatorname{new}}=Z(\mathcal{T}_{\operatorname{new}})$
and a parameter $\Theta(z_{\operatorname{new}})$. By using
$\Theta(z_{\operatorname{new}})$ as the initial value of the parameter
$\theta$, we can learn the model for the new task
$\mathcal{T}_{\operatorname{new}}$ efficiently even with a limited number of
samples for the new task.

(a) Existing

(b) Ours

Figure 5: Difference between the existing method and our method for the
transformation of a task to parameters of the model.

####  5.2.2 Generalization to $C^{*}$-algebra

We generalize the above setting to that with $C^{*}$-algebra. We construct
$f=f^{\theta}$ in Eq. (2) as a map that maps the constant function
${x}\equiv\mathbf{x}$ for a sample $\mathbf{x}$ to the constant function
${y}\equiv\mathbf{y}$ for its label $\mathbf{y}$. As described in Section
5.2.1, for a given new task $\mathcal{T}_{\operatorname{new}}$, we can get a
corresponding low-dimensional representation
$z_{\operatorname{new}}=Z(\mathcal{T}_{\operatorname{new}})$. However, in this
case, we use also vectors in the neighborhood of $z_{\operatorname{new},1}$.
We set $\mathcal{Z}$ as a neighborhood of $z_{\operatorname{new}}$ and set the
$\mathcal{A}$-valued parameter $\theta$ as a function on $\mathcal{Z}$. For
given samples $\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\in\mathbb{R}^{N_{0}}$ and
its labels $\mathbf{y}_{1},\ldots,\mathbf{y}_{n}\in\mathbb{R}^{N_{H+1}}$, we
set the loss function $L:\mathbb{R}^{N}\to\mathbb{R}_{+}$ as the cross
categorical entropy:

$L(\theta)=-\sum_{i=1}^{n}\left\langle{y}_{i},\log(f^{\theta}({x}_{i}))\right\rangle.$

By using the restriction of the function $\Theta$ on $\mathcal{Z}$ as the
initial value of the $\mathcal{A}$-parameter $\theta$, we can learn the model
for the new task $\mathcal{T}_{\operatorname{new}}$ using the information
about tasks distributed in the neighborhood of the new task in the space of
the low-dimensional representations. This makes learning the model more
efficient even with a limited number of samples for the new task.

(a) Task $1$

(b) Task $2$

Figure 6: Test accuracies of the new tasks with different values of $l$. (Each
value is an average ($\pm$ a standard deviation) over 5 independent runs with
different values of $z_{j,i}$ ($j=1,\ldots,5$, $i=2,\ldots,l$). $\mu=0.05$ for
Task 1 and $\mu=0.1$ for Task 2.)

####  5.2.3 Numerical results

We show the validity of applying our method to few-shot learning numerically.
We used the miniImageNet dataset (Vinyals et al., 2016), which is composed of
$100$ classes, each of which has $600$ images, and considered the $5$-way
$1$-shot task in the same manner as Subsection 4.2 in (Rusu et al., 2019). We
randomly split the dataset into the train data with $1$ image and the test
data with $599$ images. In this case, each task is the classification with
respect to randomly selected $5$ classes with $1$ sample in each class. We set
$f^{\theta}$ as the one-layer softmax classifier as used by Rusu et al.
(2019), and for the gradient descent method, we used Adam with learning rate
$0.001$. In this case, the number of parameters $N$ is $N_{1}N_{0}$, where
$N_{0}=640$ and $N_{1}=5$. We first learn the model proposed by Rusu et al.
(2019) and obtain maps $Z$ and $\Theta$ explained in Subsection 5.2.1. In
their model, the dimension of the low-dimensional representation is $320$ and
for $j=1,\ldots,N_{1}$, elements $N_{0}(j-1)+1\sim N_{0}j$ of $\Theta$ only
depends on elements $64(j-1)+1\sim 64j$ in the space of the low-dimensional
representation. Thus, we set $\mathcal{Z}\subseteq\mathbb{R}^{320}$. For a new
task $\mathcal{T}_{\operatorname{new}}$, we set the finite-dimensional
subspace $\mathcal{V}$ of $\mathcal{A}^{N}$ as
$\oplus_{j=1}^{N_{1}}\operatorname{Span}\\{v_{j,1},\ldots,v_{j,l}\\}^{N_{0}}$.
Here, $l\leq 10$, $v_{j,i}(z)=e^{-10\|z_{j,i}-p_{j}(z)\|^{2}}$,
$z_{j,1}=Z(\mathcal{T}_{\operatorname{new}})_{64(j-1)+1:64j}$, and $z_{j,i}$
for $i=2,\ldots,l$ are randomly drawn from the normal distribution with mean
$z_{j,1}$ and standard deviation $0.01$. Moreover, $p_{j}$ is the projection
that maps $z$ to $z_{64(j-1)+1:64j}$ and for a finite-dimensional vector $v$,
$v_{i:j}$ denotes the $j-i+1$-dimensional vector composed of elements $i\sim
j$ of $v$. We set the map ${P}:\mathcal{A}\to\mathcal{V}$ as the kernel ridge
regression in the same manner as Section 5.1.3.

For randomly selected two new tasks, we compared the accuracy of the
classification among different values of $l$. We set $l=1,7,10$ for both
tasks. For $l>1$, the output (the probability to belong to each class) is a
function on $\mathcal{Z}$. We computed the test accuracy of the output at
$z_{1}$. Fig. 6 shows the results. Note that $l=1$ corresponds to the standard
few-shot learning (with $\mathbb{R}$-valued parameters) explained in 5.2.1. We
can get higher accuracy as $l$ becomes larger. Fig. 7 shows the accuracy with
different values of the hyper parameter $\eta$ of the kernel ridge regression
$P$. In this experiment, we fixed another hyperparameter $\lambda$ as $0$. If
$\mu=0$, then our method is equivalent to the existing method. Thus, if $\mu$
is too small, the accuracy is not so high. On the other hand, if $\mu$ is
large, then the mean squared error of the kernel ridge regression becomes
large. Thus, if $\mu$ is too large, the accuracy is not so high.

(a) Task $1$

(b) Task $2$

Figure 7: Test accuracies of the new tasks with different values of $\mu$.
($l=10$ for both tasks)

###  5.3 Other applications

Although in Subsections 5.1 and 5.2 we focus on density estimation and few-
shot learning, we can apply our framework to other applications. We list
examples of other applications and discuss connections with existing methods
below.

##### Ensemble learning

Ensemble learning combines multiple models to obtain better generalization
performance (Dong et al., 2020; Ganaie et al., 2021). Our framework allows us
to combine the models continuously. Indeed, the case of
$\tilde{\lambda}=\eta=0$ and $\mathcal{D}=\sum_{i=1}^{l}\delta_{z_{i}}/l$ in
Subsection 5.1.3 is equivalent to the existing framework of ensemble learning.
In general, let $\mathcal{Z}=\\{z_{1},\ldots,z_{m}\\}$ be a finite discrete
set and let $P$ and $\tilde{\lambda}$ in Eq. (4) be the identity map and $0$,
respectively. In addition, let the loss function $L$ be defined as
$L(\theta)(z)=\tilde{L}(\theta(z),z)$. Then, our framework is reduced to the
ensemble learning because for each $i=1,\ldots,m$, $f^{\theta}(\cdot)(z_{i})$
in Eq. (2) is the classical model with $\mathbb{R}$-valued parameters, and the
learning process is independent of that for $j\neq i$. By setting
$\mathcal{Z}$ as an infinite set and $P$ as a map different from the identity
map, we can learn multiple models more efficiently.

##### Generating time-series or spatial data

We set $\mathcal{Z}\subseteq\mathbb{R}$ for time-series data and we set
$\mathcal{Z}\subseteq\mathbb{R}^{2}$ or $\mathcal{Z}\subseteq\mathbb{R}^{3}$
for spatial data. Then, we set $f=f^{\theta}$ in Eq. (2) so that
$f^{\theta}(\cdot)(z)$ is a generative model such as GAN (Goodfellow et al.,
2014; Karras et al., 2020), VAE (Kingma & Welling, 2014; Gregor et al., 2015),
or normalizing flow (Dinh et al., 2014, 2017; Teshima et al., 2020) for any
$z\in\mathcal{Z}$. Since the outputs of the model are functions on
$\mathcal{Z}$ in our framework, we can generate time-series or spatial data as
a function on $\mathcal{Z}$ rather than a discrete series.

##### Learning distributions of parameters

Distributions of parameters of models have been studied (Pennington et al.,
2018; Sonoda et al., 2021). Franchi et al. (2020) propose learning
distributions of parameters of a model rather than their values. They assume
that the distributions are normal distributions. Using our framework, we can
consider more general distributions. Let $\mathcal{Z}$ be a probability space
and we set the $\mathcal{A}$-valued parameters $\theta\in\mathcal{A}^{N}$ as
random variables taking their values in $\mathbb{R}$. If we limit the
distributions of $\theta$ to Dirac measures, then our framework is reduced to
be a classical model with $\mathbb{R}$-valued parameters. If we limit the
distributions of $\theta$ to normal distributions, then our framework is
reduced to be the framework proposed by Franchi et al. (2020).

##### Generalizing complex-valued networks

Using complex-valued variables and parameters of models for taking advantage
of the arithmetic of complex numbers has been studied (Bassey et al., 2021;
Hirose, 1992; Amin et al., 2008; Nishikawa et al., 2005; Yadav et al., 2005).
Since $C^{*}$-algebra is a generalization of the space of complex numbers, our
framework generalizes complex-valued networks. In addition, the
$C^{*}$-algebra $C(\mathcal{Z})$ is the space of complex-valued continuous
functions on $\mathcal{Z}$. Thus, by using our framework, we can aggregate
multiple complex-valued networks in the same way as real-valued networks.

##  6 Conclusion

In this paper, we proposed a new general framework of neural networks on
$C^{*}$-algebra. We focused on the $C^{*}$-algebra of the space of continuous
functions on a compact space and provided a gradient descent method for
learning the model on $C^{*}$-algebra. By generalizing the parameters of a
model to functions, we can use tools for functions such as regression and
integrations, which enables us to learn features of data efficiently and adapt
the models to problems continuously. We applied our framework to density
estimation and few-shot learning and showed the validity of our framework. Our
framework is valid for a wide range of practical applications and not limited
to the above cases.

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## Appendix

##  Appendix A Definitions and examples related to Section 3

We provide the standard definitions and examples related to Section 3.

######  Definition A.1 (Algebra).

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

$\bullet$ $(b+c)\cdot d={b}\cdot d+c\cdot d$, $\bullet$ $b\cdot(c+d)=b\cdot
c+b\cdot d$, $\bullet$ $(\alpha c)\cdot d=\alpha(c\cdot d)=c\cdot(\alpha d)$.
The symbol $\cdot$ is omitted when doing so does not cause confusion.

######  Definition A.2 (Multiplication).

Let $\mathcal{M}$ be an abelian group with operation $+$ and let $\mathcal{A}$
be a ring. 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

$\bullet$ $(u+v)\cdot c=u\cdot c+v\cdot c$, $\bullet$ $u\cdot(c+d)=u\cdot
c+u\cdot d$, $\bullet$ $u\cdot(cd)=(u\cdot{c})\cdot{d}$, $\bullet$ $u\cdot
1_{\mathcal{A}}=u$ if $\mathcal{A}$ is unital,
then $\cdot$ is called an $\mathcal{A}$-multiplication. The symbol $\cdot$ is
omitted when doing so does not cause confusion.

######  Example A.3.

A simple example of Hilbert $C^{*}$ modules over a $C^{*}$-algebra
$\mathcal{A}$ is $\mathcal{A}^{N}$ for $N\in\mathbb{N}$. The
$\mathcal{A}$-valued inner product of $\mathbf{c}=[c_{1},\ldots,c_{n}]^{T}$
and $\mathbf{d}=[d_{1},\ldots,d_{n}]^{T}$ is defined as
$\left\langle\mathbf{c},\mathbf{d}\right\rangle=\sum_{i=1}^{n}c_{i}^{*}d_{i}$.
The norm in $\mathcal{A}^{N}$ is given as
$\|\mathbf{c}\|=\|\sum_{i=1}^{n}c_{i}^{*}c_{i}\|^{1/2}$.

##  Appendix B Proof of Fact 4.4

######  Fact 3.4.

Let $\mathcal{Z}$ be a compact metric space and let $d_{\mathcal{Z}}$ be the
metric on $\mathcal{Z}$. Let $\zeta_{0},\zeta_{1},\ldots\in\mathcal{A}$ be a
sequence such that there exists a function $\zeta^{*}$ on $\mathcal{Z}$,
$\lim_{t\to\infty}\zeta_{t}(z)=\zeta^{*}(z)$ for any $z\in\mathcal{Z}$. If
$\zeta_{0},\zeta_{1},\ldots$ is uniformly Lipschitz continuous, that is, there
exists $C>0$ such that for any $t\in\mathbb{N}$,

$|\zeta_{t}(z_{1})-\zeta_{t}(z_{2})|\leq Cd_{\mathcal{Z}}(z_{1},z_{2}),$

then $\zeta_{0},\zeta_{1},\ldots$ converges uniformly to $\zeta^{*}$ and
$\zeta^{*}\in\mathcal{A}$.

###### Proof.

For $z,c\in\mathcal{Z}$ and $s,t\in\mathbb{N}$, we have

$\displaystyle|\zeta_{s}(z)-\zeta_{t}(z)|\leq|\zeta_{s}(z)-\zeta_{s}(c)|+|\zeta_{s}(c)-\zeta_{t}(c)|+|\zeta_{t}(c)-\zeta_{t}(z)|\leq
2Cd_{\mathcal{Z}}(z,c)+|\zeta_{s}(c)-\zeta_{t}(c)|,$ (8)

where the last inequality holds since $\zeta_{0},\zeta_{1},\ldots$ is
uniformly Lipschitz continuous. Let $\epsilon>0$. Since $\zeta_{t}(c)$
converges to $\zeta^{*}(c)$, there exists $T_{c}>0$ such that for any $s,t\geq
T_{c}$, $|\zeta_{s}(c)-\zeta_{t}(c)|\leq\epsilon$ holds. Let
$\mathcal{Z}_{c}=\\{z\in\mathcal{Z}\,\mid\,d_{\mathcal{Z}}(z,c)<\epsilon\\}$.
Then, we have $\mathcal{Z}=\bigcup_{c\in\mathcal{Z}}\mathcal{Z}_{c}$. Since
$\mathcal{Z}$ is compact, there exists $n\in\mathbb{N}$ and
$c_{1},\ldots,c_{n}\in\mathcal{Z}$ such that
$\mathcal{Z}=\bigcup_{i=1}^{n}\mathcal{Z}_{c_{i}}$. For
$z\in\mathcal{Z}_{c_{i}}$, by Eq. (8), we have

$\displaystyle|\zeta_{s}(z)-\zeta_{t}(z)|\leq 2C\epsilon+\epsilon$

for $s,t\geq T_{c_{i}}$. Thus, for any
$s,t\geq\max_{i\in\\{1,\ldots,n\\}}T_{c_{i}}$ and $z\in\mathcal{Z}$, we have
$|\zeta_{s}(z)-\zeta_{t}(z)|\leq(2C+1)\epsilon$, which implies the uniform
convergence of $\zeta_{t}$. Since $\mathcal{A}=C(\mathcal{Z})$ is the Banach
space equipped with the sup norm, $\zeta^{*}\in\mathcal{A}$. ∎

(a) $\mathbb{R}$-valued model

(b) $C^{*}$-algebra-valued model

Figure 8: Difference between $\mathbb{R}$-valued and $C^{*}$-algebra-valued
models.

##  Appendix C Difference between $\mathbb{R}$-valued and $C^{*}$-algebra-
valued models

We illustrate the difference between $\mathbb{R}$-valued and $C^{*}$-algebra-
valued models for a specific case of the linear regression problem on
$\mathbb{R}$. Let $z_{1},z_{2},z_{3}\in\mathcal{Z}$ be variables that
characterize models (e.g., the index of tasks and the class of data). For
$\mathbb{R}$-valued models, we separately learn three models and get
$\mathbf{f}^{\boldsymbol{\theta}}_{i}(\mathbf{x})=\boldsymbol{\theta}_{i,1}\mathbf{x}+\boldsymbol{\theta}_{i,2}$
for $i=1,2,3$ and $\mathbf{x}\in\mathbb{R}$. Here,
$\boldsymbol{\theta}_{i,j}\in\mathbb{R}$ is an $\mathbb{R}$-valued parameter.
On the other hand, for $C^{*}$-algebra-valued models, we aggregate three
models continuously and represents them as a function. Then, we learn the
model and get $f^{\theta}(x)(z)=\theta_{i,1}(z)x(z)+\theta_{i,2}(z)$ for
$i=1,2,3$ and $x\in\mathcal{A}$, which corresponds to learn the models
simultaneously with interactions.


Further author information: (E-mail<EMAIL_ADDRESS>Telephone: +34
922433642

# Characterisation and assessment of the SOXS Spectrograph UV-VIS Detector
System

Rosario Cosentino  INAF - Fundación Galileo Galilei, Breña Baja, Spain  Marcos
Hernandez  Hector Ventura  Sergio Campana  Riccardo Claudi  Pietro Schipani
Matteo Aliverti  Laura Asquini  Andrea Baruffolo  Federico Battaini  Sagi Ben-
Ami  Weizmann Institute of Science, Rehovot, Israel  Alex Bichkovsky  Giulio
Capasso  Francesco D’Alessio  Paolo D’Avanzo  Ofir Hershko  Hanindyo
Kuncarayakti  Tuorla Observatory, Department of Physics and Astronomy,
University of Turku, Turku, Finland  Marco Landoni  Matteo Munari  Giuliano
Pignata  Adam Rubin  Salvatore Scuderi  INAF - Osservatorio Astrofisico di
Catania, Catania, Italy  Fabrizio Vitali  David Young  Jani Achren  Jose
Antonio Araiza-Duran  Iair Arcavis  Anna Brucalassi  Universidad Andres Bello,
Santiago, Chile  Rachel Bruch  Enrico Cappellaro  Mirko Colapietro  Massimo
Della Valle  Rosario Di Benedetto  Simone Di Filippo  Sergio D’Orsi  Avishay
Gal-Yam  Matteo Genoni  Jari Kotilainen  Tuorla Observatory, Department of
Physics and Astronomy, University of Turku, Turku, Finland  Gianluca Li Causi
Laurent Marty  Seppo Mattila  Michael Rappaport  Kalyan Radhakrishnan  Davide
Ricci  Marco Riva  Bernardo Salasnich  Salvatore Savarese  Stephen Smartt
Ricardo Zanmar Sanchez  Maximilian Stritzinger  Matteo Accardo  Leander H.
Mehrgan  Derek Ives

###### Abstract

The SOXS spectrograph, designed for the ESO NTT telescope, operates in both
the optical (UV-VIS: 350-850 nm) and NIR (800-2000 nm) bands. This article
provides an overview of the final tests conducted on the UV-VIS camera system
using a telescope simulator. It details the system’s performance evaluation,
including key metrics such as gain, readout noise, and linearity, and
highlights the advancements made in the upgraded acquisition system. The
testing process, conducted in the Padua laboratory, involved comprehensive
simulations of the telescope environment to ensure the results closely
resemble those expected at the ESO-NTT telescope. The successful completion of
these tests confirms the system’s readiness for deployment to Chile, where it
will be installed on the NTT telescope, marking a significant milestone in the
SOXS project.

###### keywords:

spectrograph, UV-VIS, detector Control System, CCD

(Ref. 1, 2, 3, 4, 5, 6, 7).

##  1 The SOXS UV-VIS arm

The characterization and assessment of the UV-VIS detector system were
conducted utilizing SOXS UV-VIS spectrograph, installed in the telescope
simulator. The telescope simulator, located in the Padua laboratory,
incorporates a mechanical derotator and emulates the electronic environment,
including racks, harnessing, and power supply, mirroring the final
configuration anticipated for installation on the ESO-NTT telescope. The
support electronics and software environment mirror the final version of the
instrument to ensure test results be comparable with the expected outcomes at
the telescope. Various readout modes were tested, and key characteristics
including gain, readout noise, and linearity of the detector were measured.

---
Figure 1:  SOXS UV-VIS mounted in the telescope simulator.

##  2 Overview of the UV-VIS Detector System

The UV-VIS detector system comprises an e2v CCD 44-82 sensor, a custom
detector head integrated with the ESO CFC cooling system, the NGC CCD
controller engineered by ESO, and a software interface for managing data
acquisition. (Figure 3).

---
Figure 2:  The UV-VIS Detector housing (left) and The UV-VIS Spectrograph with
the Detector housing (right).

The CCD control system is critical for maintaining the operational stability
and performance of the UV-VIS detector. The e2v CCD 44-82 sensor is known for
its high quantum efficiency and low noise, which are essential for capturing
faint astronomical signals. The detector head ensures optimal thermal
management by using the ESO CFC cooling system, which minimizes dark current
and maintains the CCD at a stable, low temperature. The NGC CCD controller
provides advanced readout capabilities with multiple readout modes, allowing
for flexibility in observational strategies. It supports various gains and
readout speeds, enabling the optimization of signal-to-noise ratio (SNR) for
different observational conditions. The software interface, designed for ease
of use, allows astronomers to select image types (e.g., bias, dark, normal),
configure acquisition parameters, and monitor system performance in real-time.

Figure 3 illustrates the Detector user interface along with the results of an
acquisition displayed in a graphical user interface (GUI). Through the user
interface, it is possible to select the image type (bias, dark, normal, etc.),
the readout mode with various gains and readout speeds, and other acquisition
parameters.

---
Figure 3:  Detector user interface along with the results of an acquisition
displayed in a graphical user interface (GUI).

##  3 Testing and Performance Evaluation of the SOXS UV-VIS Spectrograph

These tests were conducted in Padua, leveraging the advanced infrastructure of
the SOXS spectrograph and NGC (Next Generation Controller) integrated into the
telescope interface (refer to Figure 2). The interaction between these
components was facilitated by the SOXS Instrument Control System (ICS),
ensuring precise and coordinated functionality throughout the testing process.
The utilization of the SOXS spectrograph and NGC within the telescope
interface provides a comprehensive simulation environment, replicating the the
interaction between the detector system and the observational setup. This
simulation extends beyond mere hardware integration, and includes the
complexity of data acquisition, processing, and control, mirroring the
operational conditions expected during actual astronomical observations. The
results obtained from these tests were compared with previous assessments
conducted at the FGG (Fondazione Galileo Galilei) laboratory, providing a
robust benchmark for performance evaluation. This comparative analysis ensures
continuity and consistency in assessing the system’s functionality and
performance across different testing environments. Furthermore, the readout
values obtained across various operational modes were documented, providing
valuable insights into the system’s behavior under different configurations.
These values serve as crucial reference points for calibrating the system and
optimizing its performance for specific observational objectives. In summary,
the comprehensive testing regimen conducted in Padua, utilizing the SOXS
spectrograph and NGC within the telescope interface and operated via the SOXS
ICS, provided valuable insights into the system’s performance characteristics.
The comparative analysis with previous assessments and detailed documentation
of readout values further enhance our understanding of the system’s
capabilities and readiness for astronomical observations.

The following sub paragraphs provide detailed information on the main
characteristics of the readout modes implemented in the system, which can be
used to choose the best configuration to meet astronomical needs under
different observing conditions.

###  3.1 Detailed Characteristics of Readout Modes for Optimal Astronomical
Configurations

This section provide detailed information on the main characteristics of the
readout modes implemented in the system, which can be used to choose the best
configuration to meet astronomical needs under different observing conditions.
Table 1 outlines the gain, readout noise, and readout time for each readout
mode, helping in the selection of the optimal setup for various astronomical
applications.

Table 1: gain, readout noise, and readout time for each readout mode

Table 2 displays the dynamic range of the CCD (Charge-Coupled Device) along
with the various gain settings that can be used. To optimize the observation
of an object, the appropriate configuration will be selected based on the
object’s brightness. This selection process ensures that the CCD captures the
best possible image by adjusting its sensitivity according to the light
intensity of the object being observed.

Table 2: CCD dynamic range covered depending on gain

###  3.2 CCD Image Readout Methods and Their Impact on Readout Speeds

The acquisition system implements various methods for reading CCD images,
including full image readout, selected box readout, and different binning
configurations of the full image. The readout speeds vary depending on the CCD
area selected for reading. Table 3 and Table 4 lists the readout speeds
depending on the CCD area selected for reading, offering insights into how
different configurations affect readout time.

Table 3:  readout time for Fast speed mode Table 4:  readout time for Med
speed mode
---
Figure 4:  Full image, Box and binnings of the CCD.

###  3.3 Linearity

Figure 5 displays the linearity and deviation from linearity of one of the
readout modes implemented in the system. The measurement of CCD linearity was
conducted by capturing images at different exposure times and plotting the
average pixel value against the exposure time. A linear regression was then
performed to determine the linearity and identify any deviations from the
expected linear response. The results show that the linearity meets the
requirements across the dynamic range of the CCD sensor, ensuring accurate and
reliable performance in various operating conditions.

---
Figure 5:  ”Linearity and deviation of the CCD across the entire dynamic
range.”.

###  3.4 Final Testing and Validation Using Real Spectra

The testing process consist of series of steps aimed at assessing the quality
of data produced by the spectrograph and CCD system. Central to this process
was the capture of calibration images, crafted to illuminate specific facets
of system performance. These calibration images serve as tools for identifying
and mitigating systematic errors or artefacts inherent to the detector system.
By analysing the characteristics of these images, astronomers can discern
subtle irregularities in the data, enabling targeted adjustments to enhance
overall data quality. Moreover, the testing process involved the extraction
and analysis of spectra, providing a direct measure of the system’s spectral
resolution and sensitivity. By extracting and scrutinizing these spectra,
astronomers can obtain valuable information about the target object’s physical
properties and chemical composition. This comprehensive analysis not only
validates the accuracy and fidelity of the acquired data but also provides
valuable insights into the system’s performance under real-world observational
conditions. The figure presented alongside showcases an illustrative example
of this testing process, juxtaposing a calibration image (Figure 6) with the
corresponding extracted spectrum (Figure 7). This visual representation offers
a compelling demonstration of the system’s capability to faithfully capture
and analyse spectral data. In summary, the testing process employed a
multifaceted approach to assess the quality of data acquired by the
spectrograph and CCD system. By capturing calibration images and extracting
spectra, we evaluated the system’s performance and ensure the accuracy and
fidelity of the acquired data. This testing regimen serves as a crucial
validation step, confirming the system’s readiness for astronomical
observations.

---
Figure 6:  ”Real spectra of UV-VIS Spectrograph on CCD (RAW image).”.
---
Figure 7:  ”Real spectra of UV-VIS Spectrograph (extracted in the green box of
Figure 6.”.

###  3.5 Conclusions

The tests conducted with the UV-VIS arm of the SOXS spectrograph mounted on
the telescope simulator and with the support electronics that will be
installed at the telescope in Chile have functioned correctly and as expected.
We measured the parameters characterizing the acquisition system, such as
gain, readout noise, and linearity, and obtained the initial images and
extracted the first spectra. The system is ready to be shipped to Chile and
installed on the NTT telescope. The successful characterization and testing of
the UV-VIS detector system confirm that it meets the stringent requirements
for astronomical observations. The measured performance metrics, including
gain, readout noise, and linearity, align with the design specifications,
ensuring high-quality data acquisition. The readiness of the system for
deployment marks a significant milestone in the SOXS project, paving the way
for groundbreaking observations at the ESO NTT telescope. In addition to
validating the core performance parameters, the final tests demonstrated the
robustness of the upgraded acquisition system. The enhanced software interface
and flexible readout options provide astronomers with versatile tools for
optimizing observations under varying conditions. The ability to emulate the
electronic and mechanical environment of the ESO-NTT telescope within the
Padua laboratory was crucial in achieving these results, ensuring that the
system’s behavior in the field will closely match the simulated conditions.
Overall, the comprehensive testing regimen, from initial calibration to
spectral extraction, has thoroughly vetted the UV-VIS arm’s capabilities. The
system’s compliance with the design requirements and its successful
performance in a controlled, simulated environment underscore its readiness
for the challenging observational tasks ahead. With the deployment to Chile
imminent, the SOXS spectrograph is poised to contribute significantly to our
understanding of the universe through precise and reliable data collection.

###### Acknowledgements.

A special acknowledgement to the European Southern Observatory for the support
provided and for the availability to share its knowledge and to allow for the
use of the ESO laboratories in Garching.

## References

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  * [2] Cosentino, R. et al., “Development status of the uv-vis detector system of SOXS for the ESO-NTT telescope,” in [Proc. SPIE 114476C ], (2020).
  * [3] Cosentino, R. et al., “Laboratory test of the vis detector system of soxs for the eso-ntt telescope,” in [Proc. SPIE 12184-207 ], (2022).
  * [4] Colapietro, M. et al., “The integration of the soxs control electronics towards the pae,” in [Proc. SPIE 13096-107 ], (2024).
  * [5] Radhakrishnand, K. et al., “Integration and verification of the soxs instrument,” in [Proc. SPIE 13096-268 ], (2024).
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