# Multiscale Hodge Scattering Networks for Data Analysis

Naoki Saito<EMAIL_ADDRESS>Stefan C. Schonsheck<EMAIL_ADDRESS>Eugene Shvarts<EMAIL_ADDRESS>
###### Abstract

We propose new scattering networks for signals measured on simplicial
complexes, which we call _Multiscale Hodge Scattering Networks_ (MHSNs). Our
construction is based on multiscale basis dictionaries on simplicial
complexes, i.e., the $\kappa$-GHWT and $\kappa$-HGLET, which we recently
developed for simplices of dimension $\kappa\in{\mathbb{N}}$ in a given
simplicial complex by generalizing the node-based Generalized Haar-Walsh
Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). The
$\kappa$-GHWT and the $\kappa$-HGLET both form redundant sets (i.e.,
dictionaries) of multiscale basis vectors and the corresponding expansion
coefficients of a given signal. Our MHSNs use a layered structure analogous to
a convolutional neural network (CNN) to cascade the moments of the modulus of
the dictionary coefficients. The resulting features are invariant to
reordering of the simplices (i.e., node permutation of the underlying graphs).
Importantly, the use of multiscale basis dictionaries in our MHSNs admits a
natural pooling operation that is akin to local pooling in CNNs, and which may
be performed either locally or per-scale. These pooling operations are harder
to define in both traditional scattering networks based on Morlet wavelets,
and geometric scattering networks based on Diffusion Wavelets. As a result, we
are able to extract a rich set of descriptive yet robust features that can be
used along with very simple machine learning methods (i.e., logistic
regression or support vector machines) to achieve high-accuracy classification
systems with far fewer number of parameters to train than most modern graph
neural networks. Finally, we demonstrate the usefulness of our MHSNs in three
distinct types of problems: signal classification, domain (i.e.,
graph/simplex) classification, and molecular dynamics prediction.

###### keywords:

Scattering transform , simplicial complexes , multiscale graph basis
dictionaries , Hodge Laplacians , Haar-Walsh wavelet packets , graph
classification , signal classification and regression

hide

organization=Department of Mathematics, University of California, Davis,
addressline=One Shields Avenue, city=Davis, postcode=95616, state=CA,
country=USA

##  1 Introduction

_Scattering Transforms_ were introduced by Mallat in [1] as a method for
feature extraction for signals and images. These features are translation
quasi-invariant, stable to deformation, and preserve high-frequency
information from the input which make them ideal for a wide variety of data
classification problems, e.g., texture image classification [2, 3]. In
addition, their computational architecture closely resembles a convolutional
neural networks (CNNs), which allows for fast, GPU-friendly computation. In
fact, these networks are often thought of as a type of CNN, with predetermined
wavelet filter banks as their convolution filters and a pointwise modulus
operation as their activation function. A key advantage of these networks over
traditional CNNs is that since the filter banks do not need to be learned from
input data, they are much less data-hungry. Additionally, they are more
interpretable since each channel in the hidden representation is a
deterministic cascade of wavelet transform convolutions with nonlinear
activation and averaging operators.

More recently, Gao et al. introduced an analogous network architecture for
node-signals on undirected graphs [4], which they named as “Geometric
Scattering (Networks).” In this context, invariance to node-permutation takes
the place of translation-in variance. This is achieved in a manner similar to
PointNet [5], by aggregating the node features through either a sum or max-
values operation into a single value for each channel. This leads to a
deformation-stable feature extractor that can be utilized in concert with a
very simple learning model (normally logistic regression or support vector
machine) to achieve near state-of-the-art classification results on many
datasets, with far fewer training examples than CNN-based approaches often
require. As a result, these networks generate descriptive yet robust features
that can be used along with very simple machine learning methods (i.e.,
support vector machines or logistic regression) to achieve high-accuracy
classification systems with only small amounts of training data and with far
fewer number of parameters to train.

In this article, we advance this line of research to apply to signals defined
on arbitrarily high-dimensional simplicial structures, i.e., edges, triangles,
pyramids, and their $\kappa$-dimensional analogous. Our methods differ from
previous work in two key ways. First, previous scattering transform networks
have applied only to point-valued signals whereas our construction generalizes
to high-dimensional structures. Second, we utilize the $\kappa$-Hierarchical
Graph Laplace Transforms ($\kappa$-HGLET) [6, 7] and $\kappa$-Generalized
Haar-Walsh Transform ($\kappa$-GHWT)[8, 7] as the wavelet banks in the
transforms. Previous work has mostly been based on Morlet wavelets for images
and Diffusion Wavelets [9] for graph-based signals. However, we find that the
bipartition tree induced by the multiscale transforms we proposed in [7]
allows us to form sparser approximations which in turn lead to more efficient
feature extraction and therefore more expressive networks. Additionally, the
multiscale structure of these bases allows us to easily define local pooling
operations which can boost the performance of scattering networks in many
applications.

###  1.1 Comparison with Related Works

There has been a growth in recent interest in studying signals defined on
edges, triangles, and higher-dimensional substructures within graph structured
data [10, 11, 12, 13, 14]. Applications in computer vision [15, 16],
statistics [17], topological data analysis [14, 18], and network analysis [19]
have benefited from the study of high-dimensional simplicial complexes.
Convolution-based simplicial neural networks have shown remarkable results in
these new domains [20]. We extend this line of research by defining scattering
networks on these higher-dimensional domains.

Scattering networks [2] were initially introduced as a tool to explain the
success of CNNs on many computer vision problems. These networks had many of
the invariant and equivariant properties that make CNNs desirable, but did not
contain any learnable ‘filters’, and instead employed a cascade of wavelet
convolutions and contractive nonlinearities. Later, Gao et al. successfully
generalized these networks to apply to graph convolutional networks [4]. Our
work further generalizes these approaches.

The main ways our MHSNs differ from Geometric Scattering Networks (GSNs) [4]
and Deep Haar Scattering Networks (DHSNs) [21] are: 1) MHSNs accept arbitrary
simplicial complexes while GSNs/DHSNs were designed for nodes only; and 2)
GSNs and DHSNs are based on the Diffusion Wavelets of Coifman and Maggioni [9]
and the Haar transform, respectively, and hence they are not based on the
hierarchical partitioning of a given graph, while MHSNs are built over the
richer HGLET/GHWT dictionaries and more amenable for analysis since they are
composed of a collection of orthonormal bases (ONBs). Hodgelets [22] use a
kernel defined in the spectral domain, similar to the spectral graph wavelet
transform [23] to define another family of wavelet-like frames for signals on
simplicial complexes. Topological Slepians [24] also form a localized basis
dictionary on a given collection of $\kappa$-simplices, but their construction
is based on the maximization of primal domain concentration of vectors subject
to the dual domain (the frequency domain) support set. However, both Hodgelets
and Topological Slepians are difficult to use for scattering transform type
representations since they are not hierarchically arranged.

Recently, Chew et al. introduced a method for _windowed_ scattering transforms
which achieve local-pooling like operations [25]. However, since the
underlying topology of the graph/complex is non-Euclidean, it may be difficult
to consistently define the local windows across multiple graphs [26, 27]. It
may be possible to use the partitioning scheme proposed in [7] for these
windows, but defining appropriate wavelet families for such a hybrid approach
requires further study.

##  2 Hodge Laplacians and Multiscale Basis Dictionaries

In this section we review some basic Hodge theory to define the Hodge
Laplacian on simplicial complexes and then summarize the construction of
multiscale basis functions on these spaces. For a more thorough introduction
into Hodge Theory see [10, 12, 16] and for a more detailed explanation of
multiscale basis dictionaries see [28, 7].

###  2.1 Simplicial Complexes and Boundary Operators

In this subsection we review concepts from algebraic topology to formally
define simplicial complexes and introduce some notions of how two simplices
can be “adjacent.” For a more thorough review, see [10, 12]. Given a vertex
(or node) set $V=\\{v_{1},\ldots,v_{n}\\}$, a _$\kappa$ -simplex_ $\sigma$ is
a $(\kappa+1)$-subset of $V$. A _face_ of $\sigma$ is a $\kappa$-subset of
$\sigma$, and so $\sigma$ has $\kappa+1$ faces. A _co-face_ of $\sigma$ is a
$(\kappa+1)$-simplex, of which $\sigma$ is a face. A _simplicial complex_ $C$
is a collection of simplices closed under subsets, where if $\sigma\in C$,
then $\alpha\subset\sigma\implies\alpha\in C$. In particular, if $\sigma\in
C$, so does each face of $\sigma$. Let
$\kappa_{\mathrm{max}}(C)\,:=\,\max\left\\{\kappa\,|\,\sigma\in C\text{ is a
$\kappa$-simplex}\right\\}$, and for each
$\kappa\in\\{0,1,\ldots,\kappa_{\mathrm{max}}\\}$, let $C_{\kappa}$ denote the
set of $\kappa$-simplices in $C$, and let $X_{\kappa}$ be the space of real-
valued functions on $C_{\kappa}$. When $\kappa>\kappa_{\mathrm{max}}$,
$C_{\kappa}=\emptyset$. We also refer to $C$ as a _$\kappa$ -complex_ to note
that $\kappa_{\mathrm{max}}(C)=\kappa$. Let a _$\kappa$ -region_ of $C$ refer
to any nonempty subset of $C_{\kappa}$.

Let $C$ be a simplicial complex, and $\sigma,\tau\in C_{\kappa}$, for some
$\kappa>0$. When $\sigma,\tau$ share a face, they are _weakly adjacent_ ,
denoted by $\sigma\sim\tau$. When $\sigma\sim\tau$, additionally they both
share a co-face, their _hull_ , denoted by $\operatorname{hl}(\sigma,\tau)$.
If $\sigma,\tau\in C$, $\sigma\sim\tau$, and
$\operatorname{hl}(\sigma,\tau)\in C$, then $\sigma,\tau$ are _strongly
adjacent_ , denoted by $\sigma\simeq\tau$. If $\sigma\sim\tau$, but
$\sigma\not\simeq\tau$ in $C$, then $\sigma,\tau$ are _$\kappa$ -adjacent_,
denoted
$\sigma\>\>\text{\stackunder[0pt]{$\sim$}{$\scriptscriptstyle\kappa$}}\>\>\tau$.
Figure 1 demonstrates these various adjacencies among simplices in a toy
$2$-complex.

Figure 1: In this small $2$-complex $C$, $e_{1}\sim e_{4}$ because they share
the face $v_{2}$, and $e_{1}\sim e_{2}$ because they share the face $v_{1}$.
Further $e_{1}\simeq e_{2}$ because their hull $t_{1}\in C$, but
$e_{1}\not\simeq e_{4}$, so that
$e_{1}\>\>\text{\stackunder[0pt]{$\sim$}{$\scriptscriptstyle 1$}}\>\>e_{4}$.
We have $t_{1}\sim t_{2}$ because they share the face $e_{3}$, and also
$t_{1}\>\>\text{\stackunder[0pt]{$\sim$}{$\scriptscriptstyle 2$}}\>\>t_{2}$.

Suppose $\sigma=\\{v_{i_{1}},\ldots,v_{i_{\kappa+1}}\\}$,
$i_{1}<\cdots<i_{\kappa+1}$, and $\alpha\subset\sigma$ is its face. Then,
$\sigma\setminus\alpha=\\{v_{i_{\ell^{*}}}\\}$ for some
$\ell^{*}\in\\{1,\ldots,\kappa+1\\}$. Define the natural parity of $\sigma$
with respect to its face $\alpha$ as
$\operatorname{nat}(\sigma,\alpha)\,:=\,(-1)^{\ell^{*}+1}$. An _oriented_
simplex $\sigma$ further has an orientation $p_{\sigma}\in\\{\pm 1\\}$ which
indicates whether its parity with its faces is the same as, or opposite to,
its natural parity. When $p_{\sigma}=+1$, we say $\sigma$ is in _natural
orientation_. For example, a directed edge $e=(v_{i},v_{j})$ for $i<j$ is in
natural orientation, while if $i>j$, $p_{e}=-1$. An oriented simplicial
complex contains at most one orientation for any given simplex.

Given an oriented simplicial complex $C$, for each
$\kappa\in\\{0,1,\ldots,\kappa_{\mathrm{max}}\\}$, the _boundary operator_ is
a linear operator ${B_{\kappa}}:X_{\kappa+1}\mapsto X_{\kappa}$, where for
$\sigma\in C_{\kappa+1}$, $\alpha\in C_{\kappa}$, the corresponding matrix
entries are
$[{B_{\kappa}}]_{\alpha\sigma}=p_{\sigma}p_{\alpha}\operatorname{nat}(\sigma,\alpha)$.
Likewise, the _coboundary operator_ for each
$\kappa\in\\{0,1,\ldots,\kappa_{\mathrm{max}}\\}$ is just
${B_{\kappa}}^{\scriptscriptstyle{\mathsf{T}}}:X_{\kappa}\rightarrow
X_{\kappa+1}$, the adjoint to ${B_{\kappa}}$. The entries of ${B_{\kappa}}$
express _relative_ orientation between simplex and face, and they are a
natural way to construct functions taking local signed averages, according to
adjacency in the simplicial complex.

###  2.2 Hodge Laplacian

The boundary operators just introduced represent _discrete differential
operators_ encoding the structure of $\kappa$-regions in a simplicial complex,
and so can be building blocks towards a spectral analysis of functions on
those regions. For analyzing functions on $\kappa$-simplices with $\kappa>0$,
we will construct operators based on the _Hodge Laplacian_ , or _$\kappa$
-Laplacian_. As in [15], the _combinatorial_ $\kappa$-Laplacian is defined for
$\kappa$-simplices as

$L_{\kappa}\,:=\,B_{\kappa-1}^{\scriptscriptstyle{\mathsf{T}}}B_{\kappa-1}+B_{\kappa}B_{\kappa}^{\scriptscriptstyle{\mathsf{T}}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (1)

Various forms of weighting and normalization are possible, with corresponding
advantages and difficulties, and different interpretations of the resulting
Hodge Laplacian’s Fiedler vector, as explored in [29, Chap. 4]. In our
numerical experiments, we choose the symmetrically normalized, weighted Hodge
Laplacian, defined as in [14], as follows. For each
$\kappa\in\\{0,1,\ldots,\kappa_{\mathrm{max}}\\}$, let $D_{\kappa}$ refer to a
diagonal matrix, whose diagonal entries contain an assignment of positive
weights to each $\kappa$-simplex in $C_{\kappa}$. One such choice, as in [14],
is to set a simplex’s weight as its degree, by taking $D_{\kappa+1}=I$,
$\operatorname{diag}(D_{\kappa})=|B_{\kappa}|\operatorname{diag}(D_{\kappa+1})$,
and
$\operatorname{diag}(D_{\kappa-1})=|B_{\kappa-1}|\operatorname{diag}(D_{\kappa})$,
where $[|M|]_{\alpha\sigma}=\left|[M]_{\alpha\sigma}\right|$. Define the
normalized boundary matrix
${\mathcal{B}}_{\kappa}\,:=\,D_{\kappa}^{-1/2}B_{\kappa}D_{\kappa+1}^{1/2}$.
Then the symmetrically normalized, weighted Hodge Laplacian is defined as

$\mathcal{L}_{\kappa}\,:=\,{\mathcal{B}}_{\kappa-1}^{\scriptscriptstyle{\mathsf{T}}}{\mathcal{B}}_{\kappa-1}+{\mathcal{B}}_{\kappa}{\mathcal{B}}_{\kappa}^{\scriptscriptstyle{\mathsf{T}}}\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (2)

Through the rest of this article, when we wish to refer to some variant of the
Hodge Laplacian calculated on a $\kappa$-region $R$, without specifying a
choice of normalization and/or weighting, we will use $\mathcal{L}(R)$. When
$R=C_{\kappa}$, we simplify to $\mathcal{L}\,:=\,\mathcal{L}(C_{\kappa})$.

###  2.3 The $\kappa$-HGLET

The _$\kappa$ -HGLET_ is a generalization of the _Hierarchical Graph Laplacian
Eigen Transform_ (HGLET) [30] from functions on the nodes of a graph, to
functions on the $\kappa$-simplices in a given simplicial complex [7]. The
HGLET, in turn, can be viewed as a generalization of the Hierarchical Block
Discrete Cosine Transform (HBDCT), which is generated by creating a
hierarchical bipartition of the signal domain and computing the DCT of the
local signal supported on each subdomain.

Let $\\{{\boldsymbol{\phi}}^{j}_{k,l}\\}$ be the set of basis vectors in the
$\kappa$-HGLET dictionary where $j$ denotes the level of the partition (with
$j=0$ being the root), $k$ indicates the partition within the level, and $l$
indexes the elements within each partition in increasing frequency. Let
$C^{j}_{k}$ refer to the $\kappa$-region consisting of the support of
partition $k$ at level (or scale) $j$, and let
$n^{j}_{k}\,:=\,\left|C^{j}_{k}\right|$. Hence $C^{0}_{0}=C_{\kappa}$ and
$n^{0}_{0}=|C_{\kappa}|\,=:\,n$.

In order to compute the transform, we first compute the complete set of
eigenvectors $\\{{\boldsymbol{\phi}}^{0}_{0,l}\\}_{l=0:n^{0}_{0}-1}$ of
$\mathcal{L}=\mathcal{L}(C_{0}^{0})$, and order them by nondecreasing
eigenvalues. We then partition $C^{0}_{0}$ into two disjoint $\kappa$-regions
$C^{1}_{0}$ and $C^{1}_{1}$ by forming the Fiedler vector of $\mathcal{L}$. We
note that: 1) one can use any other bipartition methods; and 2) bipartitioning
with the Fiedler vector in the $\kappa$-region setting requires additional
steps vs. the graph setting, because of its rather intricate behaviors; see
[7] for the details. We iterate the same procedure for $C^{1}_{0}$ and
$C^{1}_{1}$ to generate the eigenvectors
$\\{{\boldsymbol{\phi}}^{1}_{0,l}\\}_{l=0:n^{1}_{0}-1}$ and
$\\{{\boldsymbol{\phi}}^{1}_{1,l}\\}_{l=0:n^{1}_{1}-1}$. Note that
$n^{1}_{0}+n^{1}_{1}=n^{0}_{0}=n$, and that all of the elements in
$\\{{\boldsymbol{\phi}}^{1}_{0,l}\\}$ are orthogonal to those in
$\\{{\boldsymbol{\phi}}^{1}_{1,l}\\}$ since their supports are disjoint. The
set
$\\{{\boldsymbol{\phi}}^{1}_{0,l}\\}_{l=0:n^{1}_{0}-1}\cup\\{{\boldsymbol{\phi}}^{1}_{1,l}\\}_{l=0:n^{1}_{1}-1}$
form an ONB for vectors in $X_{\kappa}$. From here, we apply this process
recursively, generating an ONB for each level in the given hierarchical
bipartition tree. If the hierarchical bipartition tree terminates at every
region containing only a $\kappa$-simplex singleton, then the final level will
simply be the standard basis of ${\mathbb{R}}^{n}$. Each level of the
dictionary contains an ONB whose vectors have the support of roughly half the
size of the previous level. There are roughly $(1.5)^{n}$ possible ONBs formed
by selecting different covering sets of regions from the hierarchical
bipartition tree. We also note that the computational cost of generating the
entire dictionary is $\mathcal{O}(n^{3})$. See [7] for the actual algorithm to
generate the $\kappa$-HGLET on a given $C_{\kappa}$ and further details.

###  2.4 The $\kappa$-GHWT

This basis dictionary is based on the _Generalized Haar-Walsh Transform_
(GHWT) [8], which can itself be viewed as a generalization of the Haar-Walsh
wavelet packets [31, Sec. 8.1]. This is formed by first generating a
hierarchical bipartition tree of $C_{\kappa}$ as for the $\kappa$-HGLET. We
then work in a bottom-up manner, beginning with the finest level
$j={j_{\mathrm{max}}}$ where each region only contains a single element that
is the indicator vector of that region. We call them _scaling vectors_ and
label them as
$\\{{\boldsymbol{\psi}}^{{j_{\mathrm{max}}}}_{k,0}\\}_{k=0:n-1}$. For the next
level $j={j_{\mathrm{max}}}-1$, we first assign a constant scaling vector for
the support on each region. Then, for each region that contains two children
in the partition tree, we form a _Haar vector_ by subtracting the scaling
vector of the child element with a higher index from that of the child element
with a lower index. This procedure will form an ONB
$\\{{\boldsymbol{\psi}}^{{j_{\mathrm{max}}}-1}_{k,l}\\}_{k=0:K-1,l=0:l(k)-1}$
(where $K$ is the number of $\kappa$-regions at level ${j_{\mathrm{max}}}-1$
and $l(k)=1$ or $2$ depending on the region $k$) whose vectors have support of
at most 2. For the level $j={j_{\mathrm{max}}}-2$, we begin by computing the
scaling and Haar vectors as before. Next, for any region that contains three
or more elements, we also compute _Walsh vectors_ by adding and subtracting
the Haar vectors of the children regions. From here, we form the rest of the
dictionary recursively. A full description of this algorithm is given in [30]
for the $\kappa=0$ case and in [7] for the general case of $\kappa>0$.

Note that like the $\kappa$-HGLET, each level of the dictionary forms an ONB,
and at each level, basis vectors have the support of roughly half the size of
the parent level. These basis vectors also have the same support as the
corresponding $\kappa$-HGLET basis vectors (that is,
$\operatorname{supp}({\boldsymbol{\phi}}^{j}_{k,l})=\operatorname{supp}({\boldsymbol{\psi}}^{j}_{k,l})$
for all $j,k,l$). However, the computational cost of computing the
$\kappa$-GHWT is only $\mathcal{O}(n\log n)$.

##  3 Multiscale Hodge Scattering Transform

Let the $\kappa$-HGLET or $\kappa$-GHWT dictionary vectors be arranged as
${\boldsymbol{\Phi}}^{J}\,:=\,\left\\{\Phi^{j}\right\\}_{j=0}^{J}$ where each
$\Phi^{j}$ is an ONB at scale (or level) $j$ with $j=0$ being the finest scale
basis, composed of delta functions. _Note that this definition of the scale
parameter $j$ is the opposite of that used in the previous sections._ In
general, we have ${j_{\mathrm{max}}}+1\approx\log_{2}n+1$ different levels
given by the hierarchical bipartition scheme, but in practice, the features
extracted by large $j$ values are not very descriptive [4]. Hence, we
typically use the first $J+1(\leq{j_{\mathrm{max}}}+1)$ levels.

Let ${\boldsymbol{f}}\in X_{\kappa}$ and let $[{\boldsymbol{f}}]_{i}$ denote
${\boldsymbol{f}}$ evaluated at simplex $i$. We propose to compute the $q$th
moment of the 0th-order and 1st-order scattering coefficients:

$\\!\\!\\!\\!\\!S^{0}(q){\boldsymbol{f}}\,:=\,\frac{1}{n}\sum_{i=1}^{n}[{\boldsymbol{f}}]_{i}^{q},\,\quad
S^{1}(q,j){\boldsymbol{f}}\,:=\,\frac{1}{n}\sum_{i=1}^{n}\left|\Phi^{j}[{\boldsymbol{f}}]_{i}\right|^{q},\,\,0\leq
j\leq J;1\leq q\leq Q,$ (3)

and the 2nd-order scattering coefficients:

$S^{2}\left(q,j,j^{\prime}\right){\boldsymbol{f}}\,:=\,\frac{1}{n}\sum_{i=1}^{n}\left|\Phi^{j^{\prime}}\left|\Phi^{j}[{\boldsymbol{f}}]_{i}\right|\right|^{q},\,\,j=0\leq
j<j^{\prime}\leq J,\,\,1\leq q\leq Q.$ (4)

And higher-order moments can be computed in a similar manner:

$S^{m}\left(q,j^{(1)},\ldots,j^{(m)}\right){\boldsymbol{f}}\,:=\,\frac{1}{n}\sum_{i=1}^{n}\left|\Phi^{j^{(m)}}\left|\Phi^{j^{(m-1)}}\left|\cdots\left|\Phi^{j^{(1)}}[{\boldsymbol{f}}]_{i}\right|\cdots\right|\right|\right|^{q},$
(5)

where $j=0\leq j^{(1)}<\cdots<j^{(m)}\leq J$. However, due to the
combinatorial blow-up in the number of features at each order, it is rare to
use more than 2nd or 3rd-order features. Note that the $m$th order features
are computed by applying a multiplication with the appropriate (sparse) weight
$\Phi^{j}$ then applying a pointwise nonlinearity to $(m-1)$st-order features.
We notate the transform without or the sum or normalization actor as
$\hat{S}$. We can write the higher order features as:

$\displaystyle S^{m}\left(q,j^{(1)},\ldots,j^{(m)}\right){\boldsymbol{f}}$
$\displaystyle=S^{1}\left(q,j^{(m)}\right)\hat{S}^{m-1}\left(q,j^{(1)},\ldots,j^{(m-1)}\right){\boldsymbol{f}}=$
(6)
$\displaystyle\cdots=S^{1}\left(q,j^{(m)})\hat{S}^{1}(q,j^{(m-1)}\right)\cdots\hat{S}^{1}\left(q,j^{(1)}\right){\boldsymbol{f}}$

This behavior mimics the architecture of convolutional neural networks (with
fixed weights) and has been studied extensively [1]. However, unlike
traditional feed-forward networks, rather than only considering the features
extracted at end of the cascade of linear and nonlinear layers, we concatenate
the representation extracted at each layer for downstream analysis. In
general, we refer to the process of extracting these features as the
_Multiscale Hodge Scattering Transform_ (MHST). Later we will analyze these
features with relatively simple machine learning models such as support vector
machines (SVMs) and logistic regression models (LRMs). When working with such
models with learnable parameters in concert with these deterministic features,
we refer to it as the _Multiscale Hodge Scattering Network_ (MHSN).

###  3.1 Local Pooling

In general, we can gather all of the moments $\leq Q$ and of orders $\leq M$
to have a total of $Q\sum_{m=0}^{M}\binom{J}{m}$ features for a given signal.
The summations from $i=1$ to $i=n$ in (3)–(5) can be viewed as _global_
pooling operations. In situations where permutation invariance is not required
(i.e., all signals are defined on a fixed complex with known node ordering),
however, we can omit these sums, i.e., no pooling is done. As a result, we are
left with $Qn\sum_{m=0}^{M}\binom{J}{m}$ features for each signal. We can also
generate intermediate features between these two extremes: retain sums over
each region $k$ at level $j$ instead of not summing at all or summing all the
regions of level $j$ in (3)–(5). This can be viewed as _local_ pooling
operations and results in a tuple of features rather than a single value. This
is similar to the windowed scattering transforms recently proposed by [25],
but here we leverage the multiscale decomposition provided by the hierarchical
partition tree to avoid introducing a new user parameter to determine the
window size. In the case of these local pools, we replace the normalization
factor in the averaging operation with the number of elements in the local
pool rather than the number in the entire simplex as defined in Eq. (7).

We denote these transforms as $S^{m}_{j}$ where $j$ denotes the level at which
the sum has taken place. So $S^{m}_{j_{\mathrm{max}}}=S^{m}$ indicated the
standard max-pooling-like scheme as defined by [2, 4]. Then
$S^{m}_{0}=\hat{S}$ denotes the transform without any sums and $S^{m}_{2}$
denotes the transform with local pools determined by the third level (i.e.,
$j=2$) of the partition tree. In general we have:

$S^{m}_{j}\left(q,j^{(1)},\ldots,j^{(m)}\right){\boldsymbol{f}}\,:=\,\Biggl{\\{}\frac{1}{n^{j}_{k}}\sum_{i\in
C^{j}_{k}}\left|\Phi^{j^{(m)}}\left|\Phi^{j^{(m-1)}}\left|\cdots\left|\Phi^{j^{(1)}}[{\boldsymbol{f}}]_{i}\right|\cdots\right|\right|\right|^{q}\Biggr{\\}}_{k=0}^{K^{j}-1}$
(7)

Note that the subscript $j$ indicates where the final averaging has taken
place whereas $j^{(m)}$ indicates the index of the basis elements used to
compute the feature.

##  4 Theoretical Analysis of Multiscale Hodge Scattering Transforms

In this section we establish approximation properties of the multiscale basis
dictionaries, then use there results to detail the continuity, invariance and
equivalence properties of the MHSNs which make them desirable for signal and
domain classification problems. For notational convenience, we let $S=S^{1}$
and only consider the first order transform in our formal proofs. However,
since $S^{m}{\boldsymbol{f}}$ can be thought of as applying $S^{1}$ to the
transform $S_{0}^{m-1}{\boldsymbol{f}}$, all of the proofs can be trivially
generalized to the $S^{m}$ case.

First, for singleton $\kappa$-elements $\sigma$ and $\tau$ of $C_{\kappa}$,
and signal ${\boldsymbol{f}}$, we define a distance function and then the
associated Hölder semi-norm as the number of elements in the smallest
partition in the tree that constrains both elements, formally we have:

$d(\sigma,\tau)\,:=\,\min\left\\{n^{j}_{k}\,\Big{|}\,\sigma,\tau\in
C^{j}_{k}\right\\},\quad
C_{H}({\boldsymbol{f}})\,:=\,\sup_{\sigma\neq\tau}\frac{\left|[{\boldsymbol{f}}]_{\sigma}-[{\boldsymbol{f}}]_{\tau}\right|}{d(\sigma,\tau)^{\alpha}}$

where $\alpha$ is a constant in $(0,1]$. With these definitions, the
dictionary coefficient decay and approximation results of [32, 6] for the GHWT
and HGLET can be applied to the $\kappa$-GHWT and $\kappa$-HGLET bases as
detailed further in [7].

######  Theorem 1 (Fast Decay of Multiscale Coefficients).

For a simplicial complex $C$ equipped with a hierarchical bipartition tree,
suppose that a signal ${\boldsymbol{f}}$ is Hölder continuous with exponent
$\alpha$ and constant $C_{H}({\boldsymbol{f}})$. Then the coefficients with
$l\geq 1$ for the $\kappa$-HGLET ($c^{j}_{k,l}$) and $\kappa$-GHWT
($d^{j}_{k,l}$) satisfy:

$\left|c^{j}_{k,l}\right|\leq
C_{H}({\boldsymbol{f}})\left(n^{j}_{k}\right)^{\alpha+\frac{1}{2}},\quad\left|d^{j}_{k,l}\right|\leq
C_{H}({\boldsymbol{f}})\left(n^{j}_{k}\right)^{\alpha+\frac{1}{2}}$

###### Proof.

See Theorem 3.1 of [6]. ∎

######  Theorem 2 (Approximation Power of Multiscale Bases).

For a fixed ONB $\\{{\boldsymbol{\phi}}_{l}\\}^{n-1}_{l=0}$ and a parameter
$0<\rho<2$, then

$\|{\boldsymbol{f}}-P_{m}{\boldsymbol{f}}\|_{2}\leq\frac{|{\boldsymbol{f}}|_{\rho}}{m^{\beta}}.$

where $P_{m}$ in the best nonlinear $m$-term approximation in the basis
$\\{{\boldsymbol{\phi}}_{l}\\}^{n-1}_{l=0}$,
$\beta=\frac{1}{\rho}-\frac{1}{2}$ and $|{\boldsymbol{f}}|_{\rho}$ is defined
as
$|{\boldsymbol{f}}|_{\rho}\,:=\,\left(\sum_{l=0}^{n-1}|\langle{\boldsymbol{f}},{\boldsymbol{\phi}}_{l}\rangle|^{\rho}\right)^{\frac{1}{\rho}}$

###### Proof.

See Theorem 3.2 of [6] and Theorem 6.3 of [32]. ∎

Next we establish bounds on the coefficients generate by the multiscale basis.
Let ${\mathscr{W}}$ indicate the matrix formed by stacking a multiscale basis
into a $(J+1)n\times n$ matrix where the $j$th $n\times n$ block is the $j$th-
level ONB. Next we define a weighted inner product and the associated norm for
signals in $X_{\kappa}$ as:

$\langle{\boldsymbol{f}},\boldsymbol{g}\rangle_{w}\,:=\,\sum_{\sigma\in
C_{\kappa}}[{\boldsymbol{f}}]_{\sigma}[{\boldsymbol{g}}]_{\sigma}[{\boldsymbol{w}}]_{\sigma}\quad\text{and}\quad\|{\boldsymbol{f}}\|_{w}\,:=\,\sqrt{\langle{\boldsymbol{f}},{\boldsymbol{f}}\rangle_{w}}.$

In practice, we can often use ${\boldsymbol{w}}\equiv{\boldsymbol{1}}$, but
for some applications more exotic norms, such as those that consider the
volume of each face, may be useful.

######  Remark 1 (Tight Frames).

Let ${\boldsymbol{f}}\in X_{\kappa}$, and $\|\cdot\|_{w}$ indicate the
$\ell^{2}$-norm with respect to the metric $w$. Then we have:

$\|{\boldsymbol{f}}\|^{2}_{w}\leq\|{\mathscr{W}}{\boldsymbol{f}}\|_{w}^{2}=(J+1)\|{\boldsymbol{f}}\|^{2}_{w}$

This remark is clearly true since ${\mathscr{W}}$ is simply a collection of
$J+1$ orthogonal matrices, and orthogonal transforms preserve the
$\ell^{2}$-norm. Although trivial, this fact will be vital for later proofs.
Next we show that the MHST is a non-expansive operation. This allows for
powerful nonlinear analysis properties [33] which we detail later.

######  Theorem 3 (Non-expansive Operation).

Let $S$ be the MHST with global pooling formed by the multiscale basis
dictionary ${\mathscr{W}}$ as defined above in (3)–(5), acting on the metric
space ${\mathscr{W}}$. Then we have:

$\|S{\boldsymbol{f}}_{1}-S{\boldsymbol{f}}_{2}\|_{w}\leq\|{\boldsymbol{f}}_{1}-{\boldsymbol{f}}_{2}\|_{w}$

Moreover, let $S_{j}$ indicate the transforms with local pooling as described
in Section 3.1, then for all $\ 0\leq j\leq J$ we have:

$\|S_{j}{\boldsymbol{f}}_{1}-S_{j}{\boldsymbol{f}}_{2}\|_{w}\leq\|{\boldsymbol{f}}_{1}-{\boldsymbol{f}}_{2}\|_{w}$

###### Proof.

To show that the MHST is non-expansive, we will show that each layer is non-
expansive. Since the entire transform is defined by a cascade of these layers
(with modulus operations taken between the layers), it will also be non-
expansive. Then:

$\displaystyle
S^{1}\left(q,j\right)({\boldsymbol{f}}_{1}-{\boldsymbol{f}}_{2})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left|[\Phi^{j}({\boldsymbol{f}}_{1}-{\boldsymbol{f}}_{2})]_{i}\right|^{q}$
(8)
$\displaystyle\leq\left|\frac{1}{n}\sum_{i=1}^{n}\left|[\Phi^{j}{\boldsymbol{f}}_{1}]_{i}\right|^{q}-\frac{1}{n}\sum_{i=1}^{n}\left|[\Phi^{j}{\boldsymbol{f}}_{2}]_{i}\right|^{q}\right|=\left|S^{1}\left(q,j\right){\boldsymbol{f}}_{1}-S^{1}\left(q,j\right){\boldsymbol{f}}_{2}\right|$

Since this inequality holds for each $q$ and $j$, it is clear that taking the
$w$-norm over the whole collection of features $S^{1}{\boldsymbol{f}}$ will
also hold. Then, since each order of the transform features can be formed by
applying $S^{1}$ to the previous order we have:
$\|S{\boldsymbol{f}}_{1}-S{\boldsymbol{f}}_{2}\|_{w}\leq\|{\boldsymbol{f}}_{1}-{\boldsymbol{f}}_{2}\|_{w}$.
The proof for the local summation follows exactly as above. ∎

Next we show that our networks are invariant under group operations which
preserve the inner product $\langle\cdot,\cdot,\rangle_{w}$, such as
permutations of the $\kappa$-simplices. That is, relabeling indices of the
elements of $C_{\kappa}$ does not affect the output of the transform. For
example, given a signal ${\boldsymbol{f}}$ defined on the triangular faces of
a simplex, the indexing of the triangles does not effect the globally pooled
transform and permuting this indexing results in an equivalent permutation of
the non-pooled signal. This theorem is analogous to Theorem 3 in [25] and
Proposition 4.1 in [34], but generalizes to any $\kappa$, rather than strictly
to the nodes ($\kappa$=0 case).

######  Theorem 4 (Invariance and Equivariance).

Suppose $\mathcal{G}$ is a group of transforms on the elements of
$C_{\kappa}$(e.g., permutations of the $\kappa$-simplices). Furthermore, given
any ${\boldsymbol{\xi}}\in\mathcal{G}$, let
$V_{\boldsymbol{\xi}}:C_{\kappa}\rightarrow C_{\kappa}$ denote the operator
$V_{\boldsymbol{\xi}}{\boldsymbol{f}}={\boldsymbol{f}}\circ{\boldsymbol{\xi}}$
induced by ${\boldsymbol{\xi}}$ and $S^{[{\boldsymbol{\xi}}]}$ the analogous
transform on $C^{[{\boldsymbol{\xi}}]}_{\kappa}$, which is the permuted
version of $C_{\kappa}$. Then both of the following hold:

$S^{[{\boldsymbol{\xi}}]}V_{\boldsymbol{\xi}}{\boldsymbol{f}}=S{\boldsymbol{f}}\quad\text{
and }\quad
S_{j}^{[{\boldsymbol{\xi}}]}V_{\boldsymbol{\xi}}{\boldsymbol{f}}=S_{j}{\boldsymbol{f}}$
(9)

for all ${\boldsymbol{\xi}}\in\mathcal{G}$,
${\boldsymbol{f}}\in{\mathbb{R}}^{n}$ and $0\leq j\leq J$.

###### Proof.

As with the previous proof we will show that this holds for an arbitrary
layer, and since the entire transform is formed from a cascade of these
layers, it will also have this property. First, denote
$\mathcal{L}_{\boldsymbol{\xi}}=V_{\boldsymbol{\xi}}\circ\mathcal{L}\circ
V_{\boldsymbol{\xi}}^{-1}$ where $\mathcal{L}$ is an appropriate version of
the $\kappa$-Laplacian for $C_{\kappa}$. It is immediately clear that if
${\boldsymbol{\phi}}$ is an eigenvector of $\mathcal{L}$ with
$\mathcal{L}{\boldsymbol{\phi}}=\lambda{\boldsymbol{\phi}}$ then
$V_{\boldsymbol{\xi}}{\boldsymbol{\phi}}$ is an eigenvector of
$\mathcal{L}_{\boldsymbol{\xi}}$, i.e.,
$\mathcal{L}_{\boldsymbol{\xi}}V_{\boldsymbol{\xi}}{\boldsymbol{\phi}}=\lambda
V_{\boldsymbol{\xi}}{\boldsymbol{\phi}}$ since

$\mathcal{L}_{\boldsymbol{\xi}}V_{\boldsymbol{\xi}}=V_{\boldsymbol{\xi}}\circ\mathcal{L}\circ
V_{\boldsymbol{\xi}}^{-1}\circ
V_{\boldsymbol{\xi}}=V_{\boldsymbol{\xi}}\circ\mathcal{L}$ (10)

Then Since $\mathcal{G}$ preserves the inner product on ${\mathscr{W}}$, it is
clear that $V_{\boldsymbol{\xi}}\Phi\,:=\,\Phi_{\boldsymbol{\xi}}$ is an ONB
for ${\mathscr{W}}_{\xi}$ whose atoms are the bases $\Phi_{\xi}$ formed from
eigenvectors of $\mathcal{L}_{\boldsymbol{\xi}}$, where
$\Phi_{\xi}=V_{\xi}\Phi V_{\xi^{-1}}$

Then for arbitrary input $f$ we have:

$S^{[{\boldsymbol{\xi}}]}V_{\boldsymbol{\xi}}{\boldsymbol{f}}=\frac{1}{n}\sum_{i=1}^{n}\left|[\Phi_{\boldsymbol{\xi}}V_{\boldsymbol{\xi}}{\boldsymbol{f}}]_{i}\right|=\frac{1}{n}\sum_{i=1}^{n}\left|[V_{\boldsymbol{\xi}}\Phi{\boldsymbol{f}}]_{i}\right|=\frac{1}{n}\sum_{i=1}^{n}\left|[\Phi{\boldsymbol{f}}]_{i}\right|=S{\boldsymbol{f}}$
(11)

The proof for the local summation follows exactly as above. ∎

##  5 Signal Classification

We first demonstrate the effectiveness of our MHSNs with the article category
classification problem using the _Science News_ database [35, 36]. We do not
claim state-of-the-art results for this problem, but rather use it to
illustrate the advantages of analyzing signals via higher-dimensional
simplices. We also show how locally-pooled networks can shatter problems in
which traditional globally-pooled scattering networks fail to differentiate
between classes. The Science News dataset contains 1042 scientific news
articles classified into eight fields: Anthropology, Astronomy; Behavioral
Sciences; Earth Sciences; Life Sciences; Math/CS; Medicine; Physics. Each
article is tagged with keywords from a pool of 1133 words selected by the
database curators. We determine a simplicial complex from these keywords by
computing their word2vec [37] embeddings based on Google’s publicly available
pre-trained model [38]. We generate a symmetric $k$-nearest neighbor graph of
the embedded words and then generate $\kappa$-simplices of the graph.
Therefore, a $\kappa$-simplex in this keyword graph corresponds to
$\kappa$-face, which represents a combination of $\kappa+1$ words.

Next, we define representations of each article as a signal in each
$X_{\kappa}$ as follows. First, for $\kappa=0$ (i.e., a node-valued signal),
we define the signal ${\boldsymbol{f}}_{0}$ to be one on the nodes
representing their keywords and zero elsewhere. For $\kappa\geq 1$ we define
the signal ${\boldsymbol{f}}_{\kappa}$ to be the simplex-wise average of the
${\boldsymbol{f}}_{0}$ signal. That is,

$[{\boldsymbol{f}}_{0}]_{i}=\begin{cases}1&\text{if keyword $i$ occurs}\\\
0&\text{Otherwise}\end{cases};\quad[{\boldsymbol{f}}_{\kappa}]_{i}=\frac{1}{\kappa+1}\sum_{\begin{subarray}{c}l\in
V(\sigma_{i})\\\ \sigma_{i}\in
C_{\kappa}\end{subarray}}[{\boldsymbol{f}}_{0}]_{l},$ (12)

where $V(\sigma_{i})$ represents the set of nodes forming the $i$th simplex
$\sigma_{i}\in C_{\kappa}$. Note that these signals are highly localized since
the keywords are connected through a symmetrized $k$NN graph, and the higher-
order signals are built from the adjacency of the resulting complex. To
showcase the robustness of our approach, we report results using both $k=5$
and $k=10$ nearest neighbor graphs.

Tables 1 and 2 compare the performance of our proposed methods with the other
simpler methods, i.e., the raw expansion coefficients of the signals relative
to the standard ONBs (Delta; Fourier) or the dictionaries (Diffusion; HGLET;
GHWT). The parameters for the feature computations were set as
$(J,M,Q)=(5,3,4)$. For each $\kappa$, we performed the five-fold cross-
validation, i.e., we randomly split these 1042 signals into 80% training and
20% test sets and repeat the experiment 5 times with different train/test
splits. In every case we use the $\ell^{2}$-regularized LRM provided by
scikit-learn [39] without any additional hyperparameter tuning to compute the
classification.

Several observations are in order. First, the traditional, globally-pooled
scattering networks mostly fail on this task regardless of the wavelet
dictionary employed. Since the number of nonzero entries in each signal is
similar and therefore the $\ell^{1}$-norms are also similar, global-pooling
schemes fail to capture the keyword information (i.e., indices of nonzero
entries) in a way that differentiates between the classes and consequently do
not produce statistically significant results. The non-pooled features often
provide the highest performance, which is not surprising since there are many
more features and learnable parameters than the networks with pooling.
However, the locally-pooled features almost always perform on par with the
non-pooled features. For both the 5 and 10 nearest neighbor graphs, the best
overall results are achieved by the $\kappa$, which has the largest number of
elements. Similarly, the 10-nearest neighbor graph performs better than the
5-nearest neighbor graphs at the cost of larger $n$.

We also observe that the networks based on $\kappa$-HGLET and $\kappa$-GHWT
generally outperform those based on Diffusion Wavelets. This is likely due to
the highly localized and piecewise constant nature of the input signals, which
are well-approximated by these dictionaries [7]. In the next section, where
the signals are not localized, we do not observe this difference.

Knn = 5 | $n$ | Delta | Fourier | Diffusion | HGLET | GHWT
---|---|---|---|---|---|---
|  | Basis | Basis | Dict. | GP | NP | Dict. | GP | LP | NP | Dict. | GP | LP | NP
$\kappa$=0 | 1133 | 33.971 | 33.971 | 86.603 | 31.579 | 86.603 | 81.818 | 31.579 | 86.603 | 86.603 | 80.861 | 31.579 | 85.646 | 86.124
$\kappa$=1 | 3273 | 55.502 | 78.947 | 85.646 | 31.579 | 85.646 | 86.124 | 31.579 | 86.124 | 85.646 | 85.603 | 31.579 | 85.646 | 85.646
$\kappa$=2 | 1294 | 55.502 | 49.761 | 83.732 | 31.579 | 83.732 | 83.254 | 31.579 | 84.211 | 83.732 | 83.732 | 31.579 | 83.254 | 83.254
$\kappa$=3 | 227 | 31.579 | 31.579 | 78.947 | 31.579 | 78.947 | 51.675 | 31.579 | 78.469 | 78.947 | 51.196 | 31.579 | 78.469 | 78.947
$\kappa$=4 | 16 | 31.579 | 31.579 | 55.981 | 31.100 | 55.981 | 32.057 | 31.100 | 55.981 | 55.981 | 32.057 | 37.799 | 55.502 | 54.067

Table 1: Article category classification accuracy for $5$-NN graph of the
Science News dataset for different simplex degrees. Dict. implies that the SVM
is trained solely on the dictionary coefficients while GP, LP, NP imply
scattering networks with global, local, and no pooling, respectively

. The best performer for each $\kappa$ is indicated in bold while the
underlined bold numbers are the best among all $\kappa$’s.

Knn = 10 | $n$ | Delta | Fourier | Diffusion | HGLET | GHWT
---|---|---|---|---|---|---
|  | Basis | Basis | Dict. | GP | NP | Dict. | GP | LP | NP | Dict. | GP | LP | NP
$\kappa$=0 | 1133 | 35.238 | 35.238 | 60.952 | 32.381 | 87.619 | 81.905 | 32.381 | 88.571 | 87.619 | 80.952 | 32.381 | 87.619 | 87.619
$\kappa$=1 | 6890 | 81.905 | 81.905 | 86.667 | 32.381 | 86.667 | 85.714 | 32.381 | 89.524 | 86.667 | 85.714 | 32.381 | 89.524 | 89.524
$\kappa$=2 | 7243 | 76.19 | 76.19 | 86.667 | 32.381 | 88.571 | 85.714 | 32.381 | 88.571 | 88.571 | 88.571 | 32.381 | 89.524 | 88.571
$\kappa$=3 | 4179 | 69.524 | 69.524 | 74.286 | 33.333 | 86.667 | 86.667 | 33.333 | 86.667 | 86.667 | 86.667 | 33.333 | 86.667 | 86.667
$\kappa$=4 | 1740 | 45.714 | 45.714 | 68.571 | 35.238 | 81.905 | 73.333 | 35.238 | 81.905 | 81.905 | 81.905 | 33.333 | 81.905 | 81.905
$\kappa$=5 | 560 | 33.333 | 33.333 | 39.048 | 34.286 | 73.333 | 60.952 | 33.333 | 73.333 | 73.333 | 60.952 | 34.286 | 73.333 | 73.333
$\kappa$=6 | 98 | 32.381 | 32.381 | 32.381 | 34.286 | 62.857 | 39.048 | 35.238 | 62.857 | 62.857 | 62.857 | 35.238 | 62.857 | 60.952

Table 2: Article category classification accuracy for $5$-NN graph of the
Science News dataset for different simplex degrees. Dict. implies that the SVM
is trained solely on the dictionary coefficients while GP, LP, NP imply
scattering networks with global, local, and no pooling, respectively. The best
performer for each $\kappa$ is indicated in bold while the underlined bold
numbers are the best among all $\kappa$’s.

##  6 Domain Classification

Another vital application of geometric scattering networks and graph neural
networks (GNNs) is graph (and simplex) classification. Broadly speaking, this
problem consists of predicting a label of a social or chemical graph based on
a training set of similar graphs with different configurations (i.e.,
different numbers of nodes and edges). Since the size of the graphs varies
greatly within these datasets, we employ only the global-pooling version of
our MHSN, akin to the previous efforts reported in [4, 25], which were based
on geometric scattering methods. We employ a simple SVM with Gaussian radial
basis functions to classify the features extracted by the MHSTs. Moreover, we
tune the hyperparameters controlling the strength of the
$\ell^{2}$-regularization and the kernel coefficients via the cross-validation
scheme presented in [4] using the same search space and validation indexes.

In some applications, there is additional labeling (node labels, edge labels,
etc.), but we consider the unlabeled case for now. Therefore, we have to
compute permutation-invariant input features based only on
topological/geometric information from the boundary matrices. Since many of
the graphs are too small to contain high-degree simplices, we only consider
node and edge-based features and transforms. For the node-based signals, we
first compute the eccentricity and clustering coefficient of each node. Then,
pass each into the scattering transform and concatenate the outputs (using
global pooling). Finally, we used these features as inputs to an SVM. For the
edge-based signals, we use two sets of features: 1) the number of off-diagonal
terms in Hodge Laplacian; and 2) the average degree of the head and tail nodes
of each edge. For each problem we train three models: 1) using only node
features; 2) using only edge features; and 3) using both sets. We adopt
Gaussian radial basis functions with hyperparameter space and tuning scheme
proposed by [4].

We compare these results with those obtained by the geometric scattering
network using SVM (GS-SVM) as well as several popular GNN models including the
graph convolution network (GCN) [40], universal graph transform (UGT) [41],
dynamic graph CNN (DGCNN) [42], graph attention network (GAT) [43], and graph
feature network (GFN) [44]. For each competing method, we reproduce the
results in the significant figures reported in their original publications; we
report to 2 decimal places for our method. More information on the benchmark
datasets can be found in A. We remark that, as of this article’s writing, this
collection of networks achieves state-of-the-art results on these datasets
according to the Papers with Code Leaderboards [45]. Further details on these
datasets and their associated classification problems are presented in A and
the references therein.

Graph | Node Scattering | Edge Scattering | Combo | GS-SVM | GCN | UGT | DGCNN | GAT | GFN
---|---|---|---|---|---|---|---|---|---
Collab | 70.84 | 78.34 | 80.39 | 79.94 | 79.00 | 77.84 | 73.76 | 75.80 | 81.50
DD | 60.67 | 68.73 | 72.71 | - | - | 80.23 | 79.37 | - | 79.37
IMDB-B | 72.70 | 70.60 | 73.10 | 71.20 | 74.00 | 77.04 | 70.03 | 70.50 | 73.40
IMDB-M | 44.40 | 47.13 | 49.68 | 48.73 | 51.90 | 53.60 | 47.83 | 47.8 | 51.80
MUTAG | 85.78 | 86.31 | 85.78 | 83.50 | 85.60 | 80.23 | 79.37 | 89.40 | 85.83
PROTEINS | 73.57 | 73.04 | 75.35 | 74.11 | 76.00 | 78.53 | 75.54 | 74.70 | 76.46
PTC | 62.85 | 67.71 | 68.28 | 63.94 | 64.20 | 69.63 | 58.59 | 66.70 | 66.60

Table 3: Graph classification accuracy on seven datasets. The best performer
for each dataset is indicated in bold.

Although our MHSNs do not achieve state-of-the-art results on these datasets,
they are very competitive with _only a small fraction of the learnable
parameters_. Moreover, the number of learnable parameters in our models is not
tied to the graph size and depends only on the order of the scattering $M$ and
the number of moments $Q$ computed. For example, Table 4 compares our methods
with the UGT and the GFN, which are the state-of-the-art methods for various
graph classification problems. These methods each require more than half a
million parameters for some cases (867K for UGT) to achieve results similar to
ours, requiring only 256 parameters to learn. As a result, our MHSNs can be
implemented and trained on a consumer-level laptop, whereas many of these
competing GNNs require specialized hardware.

| Hodge Scattering + SVM | UGT | GFN
---|---|---|---
Graph | Accuracy | # Parameters | Accuracy | # Parameters | Accuracy | # Parameters
Collab | 80.39 | 256 | 77.84 | 866,746 | 81.50 | 68,754
DD | 72.71 | 256 | 80.23 | 76,928 | 79.37 | 68,754
IMDB-B | 73.10 | 256 | 77.04 | 55,508 | 73.40 | 68,754
IMDB-M | 49.68 | 256 | 53.60 | 48,698 | 51.80 | 68,818
MUTAG | 85.78 | 256 | 80.23 | 4,178 | 85.83 | 65,618
PROTEINS | 75.35 | 256 | 78.53 | 1,878 | 76.46 | 65,618
PTC | 68.28 | 256 | 69.63 | 12,038 | 66.60 | 65,618

Table 4: Comparison of MHSN and state of the art graph classification networks
in accuracy and number of learnable parameters

##  7 Molecular Dynamics

Our MHSNs can also be used for regression problems where the goal is to
predict a continuous property of a simplicial complex (or simply a graph)
based on a set of observations of the complex under various conditions.
Therefore, they are quite suitable for learning molecular dynamics,
particularly the potential energy surface of a molecule, given a few
registrations of the molecule and its energies. The Revised Molecular Dynamics
17 (rMD17 dataset) [46] contains 100,000 structures and associated energies of
various molecules. However, these structures are taken from a molecular
dynamics simulation, i.e., time series data, which is not independent and
identically distributed. To overcome this, instead of using the entire
dataset, we use a set of five standardized 80/20 training/test indexes of
1,000 snapshots of the structures/energies per molecule selected by the
authors of the dataset [46].

We extract a rich set of features for each structure (i.e., a pose or
confirmation of a molecule) using our MHSNs and then employ a support vector
regression (SVR) method with Gaussian radial basis functions to approximate
the energy. More specifically, for each molecule, we first compute the average
position of each atom across the training set. Then, using these positions, we
create a $k$NN-graph (with $k=5$) as a template and compute MHST coefficients
for nodes and edges of this simplicial complex. By using this graph, rather
than the molecular-bond graph, we can better capture the geometric information
in the pose of the molecule as detailed in [47, 48]. Then, for each pose, we
extract a set of features based only on the coordinates of the confirmation.
For the nodes-based networks, we compute the Euclidean distance matrix of the
node coordinates and then use each column as an input to the MHSN. For the
edge-based networks, we use a diagonal matrix of the edge lengths instead. We
do not use higher-degree simplices because the molecules are too small.

Table 5 shows our results for aspirin (21 atoms) and paracetamol (20 atoms)
molecules. We compare our MHSNs with several state-of-the-art GNN approaches
designed specifically for processing molecular dynamics, including SchNet
[47], PaiNN [49], and two variants of special orthogonal networks [50, 48]. We
report both the mean average error (MAE) and root mean square error (RMSE) of
the energy prediction, which are the standard metrics in the computational
chemistry literature. Our MHSNs perform competitively with these approaches
while employing roughly 1% as many learnable parameters as the competing
methods. Additionally, we again observe that our edge-based analysis
outperforms the node-based analysis. This demonstrates that higher-dimensional
simplex analysis can be more powerful than node-only approaches, even in cases
where the underlying graph may not have many higher-dimensional structures

| Diff+SVR | HGLET+SVR | GHWT+SVR | SchNet | PaiNN | SO3Net I | SO3Net II
---|---|---|---|---|---|---|---
Feature Type | Node | Edge | Both | Node | Edge | Both | Node | Edge | Both
Aspirin
MAE | 4.856 | 3.132 | 3.267 | 4.884 | 3.135 | 3.285 | 4.928 | 3.075 | 3.225 | 13.5 | 3.8 | 3.8 | 2.6
RMSE | 6.181 | 4.144 | 4.314 | 6.215 | 4.129 | 4.407 | 6.213 | 4.123 | 4.316 | 18.3 | 5.9 | 5.7 | 3.8
# Parameters | 924 | 3784 | 4708 | 924 | 3784 | 4708 | 924 | 3784 | 4708 |  $\sim$ 432k |  $\sim$ 341k |  $\sim$ 283k |  $\sim$ 341k
Paracetamol
MAE | 4.609 | 2.715 | 2.795 | 4.723 | 2.643 | 2.710 | 4.748 | 2.624 | 2.699 | 8.4 | 2.1 | 2.2 | 1.4
RMSE | 5.860 | 3.418 | 4.116 | 5.964 | 3.338 | 3.424 | 5.961 | 3.299 | 3.408 | 11.2 | 2.9 | 3.0 | 1.9
# Parameters | 924 | 3784 | 4444 | 924 | 3784 | 4444 | 924 | 3784 | 4444 |  $\sim$432k |  $\sim$ 341k |  $\sim$283k |  $\sim$341k

Table 5: Comparison of the performance of our MHSNs and the other state-of-
the-art GNNs for nuclear energy prediction. We report the accuracy via MAE and
RMSE as-well as the number of trainable parameters in each network

##  8 Conclusion

In this article, we proposed the _Multiscale Hodge Scattering
Transforms/Networks_ (MHSTs/MHSNs) for robust feature extraction from signals
on simplicial complexes that can be used in classification and regression
problems, fully utilizing our multiscale basis dictionaries on such simplicial
complexes, i.e., $\kappa$-HGLET and $\kappa$-GHWT dictionaries. Our
MHSTs/MHSNs also have pooling options for the scattering transform outputs:
no-
pooling; local-pooling; and global-pooling. Such options allow our tools to
apply for various classification and regression problems on simplicial
complexes ranging from classification of signals recorded on simplicial
complexes to classification of type of simplicial complexes (i.e., domain
classification problems) to regression of molecular energies in dynamical
systems. We demonstrated that MHSNs provide comparable results with those by
the state-of-the-art GNNs with up to a two-order of magnitude reduction in
number of learnable parameters. We strongly believe that our success here
comes from the structure and organization of our multiscale basis dictionaries
that are conveniently arranged in terms of scales and locations, which are
suitable and quite amenable for generating scattering transform coefficients.

We plan to investigate how we can interpret the MHST coefficients that are
identified as important by classification methods such as the LRMs. Because of
the nonlinearities used in the MHSTs, converting the MHST coefficients to the
features in the primal/original domain is difficult. Along this direction, we
plan to examine the optimization method proposed by Weber [51, Chap. 4], which
synthesizes an input signal that generates the significant scattering
transform coefficients at the specified coefficient indices. In a related line
of research, we will explore how the MHST coefficients can be used to identify
import relationships within graphs that can be used to narrow the training
space of various attention mechanisms/graph transformers for large-scale
problems.

## Acknowledgments

This research was partially supported by the US National Science Foundation
grants DMS-1912747 and CCF-1934568 as well as the US Office of Naval Research
grant N00014-20-1-2381. N. S. also thanks Stéphane Mallat and Gabriel Peyré
for their hospitality and support during his sabbatical stay (Oct. 2021–Jan.
2022) at the Centre Sciences des Données, Ecole Normale Supérieure (Ulm).

##  Appendix A Description of Domain Classification Datasets

#### Google Collab

[52] A collaboration dataset of 5K scientific papers represented as graphs.
The goal is to predict which of three subfields a paper belongs to given its
authors. The average number of nodes of these graphs is 45 while the number of
nodes in the smallest graph is 32 and that of the largest graph is 492.

#### DD

[53] A dataset of 1178 molecular graphs of medium to large size. The goal of
the classification problem is to predict whether the protein is enzyme or not.
The average number of nodes is 285 while the number of nodes of the smallest
graph is 30 and that of the largest graph is 5748.

#### IMDB-B and IMDB-M

[52] A pair of two datasets representing collaboration within popular movies
from the Internet Movie Database with 1K (IMDB-B) and 1.5k (IMDB-M) graphs.
Each graph represents members of cast and the goal is to predict the genre
(Action/Romance for IMDB-B and Comedy/Romance/Sci-Fi of IMDB-M). The average,
the smallest, and the largest number of nodes in IMDB-B are 20, 12, and 136
while those in IMDB-M are 13, 7, and 89.

#### MUTAG

[54] A dataset of 188 molecular graphs representing mutagenic aromatic and
heteroaromatic nitro compound. The dataset contains node-based features
(detailing atom type), but we use only the adjacency matrix for our
classification tests as in [4]. The binary classification goal is to predict
whether or not a molecule (graph) has a mutagenic effect on bacterium. The
average number of nodes of these graphs is 18 while the number of nodes of the
smallest graph is 10 and that of the largest graph is 28.

#### Proteins

[55] A dataset of 1113 molecular graphs. The goal of the classification
problem is to predict whether the protein is enzyme or not. The task here is
the same as in the DD dataset, but the molecules are both smaller and more
similar to each other (see [55] for details). The average number of nodes is
39 while the number of nodes of the smallest graph is 4 and that of the
largest graph is 620.

#### PTC

[56] A dataset of 344 molecular graphs. The goal is to predict whether or not
the compound is known to be carcinogenic in rats. The average number of nodes
of these graphs is 26 while the number of nodes of the smallest graph is 6 and
that of the largest graph is 109.

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# Complementary inequalities to Davis-Choi-Jensen’s inequality and operator
power means

A. G. Ghazanfari  Department of Mathematics, Lorestan University, P. O. Box
465, Khoramabad, Iran<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be an operator convex function on $(0,\infty)$, and $\Phi$ be a unital
positive linear maps on $B(H)$. we give a complementary inequality to Davis-
Choi-Jensen’s inequality as follows

$f(\Phi(A))\geq\frac{4R(A,B)}{(1+R(A,B))^{2}}\Phi(f(A)),$

where $R(A,B)=\max\\{r(A^{-1}B),r(B^{-1}A)\\}$ and $r(A)$ is the spectral
radius of $A$. We investigate the complementary inequalities related to the
operator power means and the Karcher means via unital positive linear maps,
and obtain the following result: If $A_{1}$, $A_{2}$,…, $A_{n}$, are positive
definite operators in $B(H)$, and $0<m_{i}\leq A_{i}\leq M_{i}$, then

$\Lambda(\omega;\Phi(\mathbb{A}))\geq\Phi(\Lambda(\omega;\mathbb{A}))\geq\frac{4\hbar}{(1+\hbar)^{2}}~{}\Lambda(\omega;\Phi(\mathbb{A})),$

where $\hbar=\max\limits_{1\leq i\leq n}\frac{M_{i}}{m_{i}}$. Finally, we
prove that if $G(A_{1},\dots,A_{n})$ is the generalized geometric mean defined
by Ando-Li-Mathias for $n$ positive definite operators, then

$\displaystyle\Phi(G(A_{1},\dots,A_{n}))\geq\left(\frac{2h^{\frac{1}{2}}}{1+h}\right)^{n-1}G(\Phi(A_{1}),\dots,\Phi(A_{n})),$

where $h=\max\limits_{1\leq i,j\leq n}R(A_{i},A_{j})$.

###### Key words and phrases:

Karcher mean; Geometric mean; Positive linear map

###### 2010 Mathematics Subject Classification:

47A63, 47A64, 47B65

##  1\. Introduction

Let $B(H)$ denote the set of all bounded linear operators on a complex Hilbert
space $H$. An operator $A\in B(H)$ is positive definite (resp. positive semi-
definite) if $\langle Ax,x\rangle>0$ (resp. $\langle Ax,x\rangle\geq 0)$ holds
for all non‐zero $x\in H$ . If $A$ is positive semi‐definite, we denote $A\geq
0$. Let $\mathcal{P}\mathcal{S},\mathcal{P}\subset B(H)$ be the sets of all
positive semi-definite operators and positive definite operators,
respectively. To reach inequalities for bounded self-adjoint operators on
Hilbert space, we shall use the following monotonicity property for operator
functions:
If $X\in B(H)$ is self adjoint with a spectrum $Sp(X)$, and $f,g$ are
continuous real valued functions on an interval containing $Sp(X)$, then

$f(t)\geq g(t),~{}t\in Sp(X)\Rightarrow~{}f(X)\geq g(X).$ (1.1)

For more details about this property, the reader is referred to [18].

For $A,B\in\mathcal{P}$ the geometric mean $A\sharp B$ of $A$ and $B$ is
defined by $A\sharp
B=A^{\frac{1}{2}}(A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{\frac{1}{2}}A^{\frac{1}{2}}.$

For $A,B\in\mathcal{P}$, let

$R(A,B)=\max\\{r(A^{-1}B),r(B^{-1}A)\\}$

where $r(A)$ means the spectral radius of $A$ and we have

$r(B^{-1}A)=\inf\\{\lambda>0:A\leq\lambda
B\\}=\|B^{\frac{-1}{2}}AB^{\frac{-1}{2}}\|.$

$R(A,B)$ was defined in [3], and many nice properties of $R(A,B)$ were shown
as follows: If $A,B,C\in\mathcal{P}$, then

$\displaystyle(i)~{}R(A,C)\leq R(A,B)R(B,C)\text{ (triangle inequality) }$
$\displaystyle(ii)~{}R(A,B)\geq 1,\text{ and }R(A,B)=1\text{ iff }A=B$
$\displaystyle(iii)~{}\|A-B\|\leq(R(A,B)-1)\|A\|.$

The Thompson distance $d(A,B)$ on the convex cone $\Omega$ of positive
definite operators is defined by

$d(A,B)=\log R(A,B)=\max\\{\log r(A^{-1}B),\log r(B^{-1}A)\\},$

see [3, 7, 16]. we know that $\Omega$ is a complete metric space with respect
to this metric and the corresponding metric topology on $\Omega$ agree with
the relative norm topology.

As a basic inequality with respect to the metric, the following inequality for
a weighted geometric mean of two operators hold [3, 7]:

$d(A_{1}\sharp_{\nu}A_{2},B_{1}\sharp_{\nu}B_{2})\leq(1-\nu)d(A_{1},B_{1})+\nu
d(A_{2},B_{2})$

for $A_{1},A_{2},B_{1},B_{2}\in\Omega$ and $\nu\in(0,1)$.

Let $A$ and $B$ be two positive definite operators on a Hilbert space $H$ and
$\Phi$ be a unital positive linear map on $B(H)$. Ando [1, Theorem 3] showed
the following property of a positive linear map in connection with the
operator geometric mean.

$\Phi(A\sharp B)\leq\Phi(A)\sharp\Phi(B).$ (1.2)

Inequality (1.2) is extended to an operator mean $\sigma$ in Kubo-Ando theory
as follows:

$\Phi(A\sigma B)\leq\Phi(A)\sigma\Phi(B),$

In particular for the weighted geometric mean, we have

$\Phi(A\sharp_{\nu}B)\leq\Phi(A)\sharp_{\nu}\Phi(B),$ (1.3)

where $\nu$ is a real number in $(0,1]$. A complementary inequality to
inequality (1.3), is the following important inequality[14]:
Let $0<m_{1}I\leq A\leq M_{1}I$ and $0<m_{2}I\leq B\leq M_{2}I$ and $0<\nu\leq
1$, then

$K(h,\nu)\Phi(A)\sharp_{\nu}\Phi(B)\leq\Phi(A\sharp_{\nu}B),$ (1.4)

where $h=\frac{M_{1}M_{2}}{m_{1}m_{2}}$ and $K(h,\nu)$ is the generalization
Kantorovich constant defined by

$K(h,\nu)=\frac{h^{\nu}-h}{(\nu-1)(h-1)}\left(\frac{(\nu-1)(h^{\nu}-1)}{\nu(h^{\nu}-h)}\right)^{\nu}.$

The special case, $K(h,2)=K(h,-1)=\frac{(1+h)^{2}}{4h}$ is called the
Kantorovich constant. The generalization Kantorovich constant $K(h,\nu)$ has
the following properties:

$\displaystyle K(h,\nu)=K(h,1-\nu)$ $\displaystyle 0$
$\displaystyle<K(h,\nu)\leq 1\text{ for }0<\nu\leq 1,$

and $K(h,\nu)$ is decreasing for $\nu\leq\frac{1}{2}$ and increasing for
$\nu>\frac{1}{2}$ therefore for all $\nu\in\mathbb{R}$,
$K(h)=K(h,\frac{1}{2})=\frac{2h^{\frac{1}{4}}}{1+h^{\frac{1}{2}}}\leq
K(h,\nu)$.

For some fundamental results on complementary inequalities to famed
inequalities, we would like to refer the readers to [14, 18]. Afterwards,
several complementary inequalities to celebrated inequalities was discussed by
many mathematicians. For more information and some recent results on this
topic; see [3, 8, 19, 20, 21].

##  2\. A complementary inequality to weighted geometric mean

First, we state another complementary inequality to (1.3) with respect to
$R(A,B)$ as follows:

######  Theorem 1.

Let $A$ and $B$ be two positive definite operators in $\mathcal{P}$, then

$K\left(R(A,B)^{2},\nu\right)\Phi(A)\sharp_{\nu}\Phi(B)\leq\Phi(A\sharp_{\nu}B).$
(2.1)

###### Proof.

we know that

$\dfrac{1}{R(A,B)}A\leq B\leq R(A,B)A.$ (2.2)

Define a linear map $\Psi$ on $H$ by

$\Psi(X)=\Phi(A)^{-\frac{1}{2}}\Phi(A^{\frac{1}{2}}XA^{\frac{1}{2}})\Phi(A)^{-\frac{1}{2}}.$

Then $\Psi$ is a unital positive linear map. From (2.2)

$\displaystyle m=\dfrac{1}{R(A,B)}\leq A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\leq
R(A,B)=M.$ (2.3)

Using (1.4) for $\Psi$, and (2.3), we get

$K(h,\nu)\Psi(I)\sharp_{\nu}\Psi(X)\leq\Psi(I\sharp_{\nu}X)$ (2.4)

where $X=A^{-\frac{1}{2}}BA^{-\frac{1}{2}}$ and
$h=\dfrac{{}_{M}}{m}=R^{2}(A,B)$.

From (2.4), we have

$K(h,\nu)\Psi(X)^{\nu}\leq\Psi(X^{\nu})$

or

$\displaystyle
K(h,\nu)\Big{(}\Phi(A)^{-\frac{1}{2}}\Phi(B)\Phi(A)^{-\frac{1}{2}}\Big{)}^{\nu}\leq\Phi(A)^{-\frac{1}{2}}\Phi\Big{(}A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\nu}A^{\frac{1}{2}}\Big{)}\Phi(A)^{-\frac{1}{2}}$
$\displaystyle
K(h,\nu)\Phi(A)^{\frac{1}{2}}\Big{(}\Phi(A)^{-\frac{1}{2}}\Phi(B)\Phi(A)^{-\frac{1}{2}}\Big{)}^{\nu}\Phi(A)^{\frac{1}{2}}\leq\Phi\Big{(}A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\nu}A^{\frac{1}{2}}\Big{)}.$

Therefore, we obtain the desired inequality (2.1). ∎

The inequality (1.4) with $\nu=\frac{1}{2}$, becomes the following inequality.

$\frac{2R(A,B)^{\frac{1}{2}}}{1+R(A,B)}\Phi(A)\sharp\Phi(B)\leq\Phi(A\sharp
B).$ (2.5)

Next, we recall the Kadison’s Schwarz inequalities

$\Phi(A^{2})\geq\Phi(A)^{2},~{}~{}\Phi(A^{-1})\geq\Phi(A)^{-1}$ (2.6)

and two complementary inequalities to them, whenever $0<m\leq A\leq M$:

$\displaystyle\frac{(m+M)^{2}}{4mM}\Phi(A)^{2}\geq\Phi(A^{2}),~{}~{}\frac{(m+M)^{2}}{4mM}\Phi(A)^{-1}\geq\Phi(A^{-1}).$
(2.7)

The following inequality unifies Kadison’s Schwarz inequalities into a single
form.

$\Phi(BA^{-1}B)\geq\Phi(B)\Phi(A)^{-1}\Phi(B).$ (2.8)

If $0<m\leq A,B\leq M$, then the following inequality is a complementary
inequality to (2.8)

$\frac{(m+M)^{2}}{4mM}\Phi(B)\Phi(A)^{-1}\Phi(B)\geq\Phi(BA^{-1}B).$ (2.9)

The similar to the proof of Theorem1, we give another complementary inequality
to (2.8) with respect to $R(A,B)$ as follows:

$\frac{(1+R(A,B)^{2})^{2}}{4R(A,B)}\Phi(B)\Phi(A)^{-1}\Phi(B)\geq\Phi(BA^{-1}B).$
(2.10)

To compare the inequalities (1.4) with (2.5) and (2.9) with (2.10), the
following examples show that neither Kantorovich constants in (1.4) and (2.9)
nor Kantorovich constants in (2.5) and (2.10) are uniformly better than the
other.

######  Example 1.

Let $A=\begin{bmatrix}2&0\\\ 0&\frac{1}{3}\\\ \end{bmatrix}$ and
$B=\begin{bmatrix}4&0\\\ 0&\frac{1}{2}\\\ \end{bmatrix}$. Clearly,
$m_{1}I=\frac{1}{3}I\leq A\leq 2I=M_{1}I$ and $m_{2}I=\frac{1}{2}I\leq B\leq
4I=M_{2}I$. Then $A^{-1}B=\begin{bmatrix}2&0\\\ 0&\frac{3}{2}\\\
\end{bmatrix}$ and $B^{-1}A=\begin{bmatrix}\frac{1}{2}&0\\\ 0&\frac{2}{3}\\\
\end{bmatrix}$, therefore $R(A,B)^{2}=4\leq
h=\frac{M_{1}M_{2}}{m_{1}m_{2}}=48$. Consequently,
$K(R(A,B)^{2},\frac{1}{2})\geq K(h,\frac{1}{2})$ and $K(R(A,B)^{2},2)\leq
K(h,2)$.

######  Example 2.

Let $C=\begin{bmatrix}2&1\\\ 1&1\\\ \end{bmatrix}$ and
$D=\begin{bmatrix}1&0\\\ 0&2\\\ \end{bmatrix}$. Clearly,
$m_{1}I=\frac{3-\sqrt{5}}{2}I\leq C\leq\frac{3+\sqrt{5}}{2}I=M_{1}I$ and
$m_{2}I=I\leq D\leq 2I=M_{2}I$. Then $C^{-1}D=\begin{bmatrix}1&-2\\\ -1&4\\\
\end{bmatrix}$ and $D^{-1}C=\begin{bmatrix}2&1\\\ \frac{1}{2}&\frac{1}{2}\\\
\end{bmatrix}$, therefore
$R(C,D)^{2}=\left(\frac{5+\sqrt{17}}{2}\right)^{2}\geq
h=\frac{M_{1}M_{2}}{m_{1}m_{2}}=\frac{(3+\sqrt{5})^{2}}{2}$. Consequently,
$K(R(C,D)^{2},\frac{1}{2})\leq K(h,\frac{1}{2})$ and $K(R(C,D)^{2},2)\geq
K(h,2)$.

######  Theorem 2.

If $f$ is an operator convex function on $(0,\infty)$ and $\Phi$ is a unital
positive linear map on $B(H)$. Then

$\displaystyle K(h,2)f(\Phi(A))\geq\Phi(f(A))\geq f(\Phi(A))$ (2.11)

for every positive definite operators $A$, where $h=R(A,B)^{2}$, or
$h=\frac{M}{n}$ whenever $0<m\leq A\leq M$.

###### Proof.

It is known that every operator convex function $f$ on $(0,\infty)$ has a
special integral representation as follows:

$\displaystyle f(t)=\alpha+\beta t+\gamma t^{2}+\int_{0}^{\infty}\frac{\lambda
t^{2}}{\lambda+t}d\mu(\lambda),$ (2.12)

where $\alpha,\beta$ are real numbers, $\gamma\geq 0$, and $\mu$ is a positive
finite measure. Thus

$\displaystyle\Phi(f(A))=\alpha
1_{K}+\beta\Phi(A)+\gamma\Phi(A^{2})+\int_{0}^{\infty}\Phi(\lambda
A^{2}(\lambda+A)^{-1})d\mu(\lambda),$

and

$\displaystyle f(\Phi(A))=\alpha
1_{K}+\beta\Phi(A)+\gamma\Phi(A)^{2}+\int_{0}^{\infty}\lambda\Phi(A)^{2}(\lambda+\Phi(A))^{-1}d\mu(\lambda).$

By (2.8), we have

$\displaystyle\Phi(\lambda A^{2}(\lambda+A)^{-1})$
$\displaystyle=\lambda\Phi(A^{2}(\lambda+A)^{-1})=\lambda\Phi(A(\lambda+A)^{-1}A)$
$\displaystyle\geq\lambda\Phi(A)(\Phi(\lambda+A))^{-1}\Phi(A)=\lambda\Phi(A)(\lambda+\Phi(A))^{-1}\Phi(A)$
$\displaystyle=\lambda\Phi(A)^{2}(\lambda+\Phi(A))^{-1}.$

Therefore $\Phi(f(A))\geq f(\Phi(A))$, since $\Phi(A^{2})\geq\Phi(A)^{2}$.

On the other hand, from (2.9) or (2.10), we get

$\displaystyle\Phi(\lambda
A^{2}(\lambda+A)^{-1})=\lambda\Phi(A^{2}(\lambda+A)^{-1})=\lambda\Phi(A(\lambda+A)^{-1}A)$
$\displaystyle\leq
K(h,2)\lambda\Phi(A)(\Phi(\lambda+A))^{-1}\Phi(A)=K(h,2)\lambda\Phi(A)(\lambda+\Phi(A))^{-1}\Phi(A)$
$\displaystyle=K(h,2)\lambda\Phi(A)^{2}(\lambda+\Phi(A))^{-1}.$

Therefore $\Phi(f(A))\leq K(h,2)f(\Phi(A))$, since $\Phi(A^{2})\leq
K(h,2)\Phi(A)^{2}$. ∎

##  3\. The power means and the Karcher means

The Karcher mean, also called the Riemannian mean, has long been of interest
in the field of differential geometry. Recently it has been used in a diverse
variety of settings: diffusion tensors in medical imaging and radar,
covariance matrices in statistics, kernels in machine learning and elasticity.
Power means for positive definite matrices and operators have been introduced
in [11, 12]. It is shown in [5, 10] that the Karcher mean and power means
satisfy all ten properties stated in [2]. The Karcher mean and power means
have recently become an important tool for studying of positive definite
operators and an interesting subject for matrix analysts and operator
theorists. We would like to refer the reader to [10, 11, 12, 13, 15, 17, 21]
and references therein for more information.

Geometric and power means of two operators can be extended over more than
$3$‐operators via the solution of operator equations as follows. Let $n$ be a
natural number, and let $\triangle_{n}$ be a set of all probability vectors,
i.e.,

$\triangle_{n}=\\{\omega=(w_{1},\dots,w_{n})\in(0,1)^{n}|\sum_{i=1}^{n}w_{i}=1\\}.$

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$ and
$\omega=(\mathrm{w}_{1},\dots,w_{n})\in\triangle_{n}$. Then the weighted
Karcher mean $\Lambda(\omega;\mathbb{A})$ is defined by a unique positive
solution of the following operator equation;

$\sum_{i=1}^{n}\mathrm{w}_{i}\log(X^{\frac{-1}{2}}A_{i}X^{\frac{-1}{2}})=0.$
(3.1)

The weighted power mean $P_{t}(\omega;\mathbb{A})$ is defined by a unique
positive solution of the following operator equation;

$I=\sum_{i=1}^{n}\mathrm{w}_{i}(X^{\frac{-1}{2}}A_{i}X^{\frac{-1}{2}})^{t}\text{
for }t\in(0,1],$

or equivalently

$X=\sum_{i=1}^{n}\mathrm{w}_{i}(X\sharp_{t}A_{i})\text{ for }t\in(0,1].$ (3.2)

For $t\in[-1,0)$, it is defined by

$P_{t}(\omega;\mathbb{A})=(P_{-t}(\omega;\mathbb{A}^{-1}))^{-1},$

where $\mathbb{A}^{-1}=(A_{1}^{-1},\dots,A_{n}^{-1})$.

Lawson and Lim in [11, Corollary 6.7] gave an important connection between the
weighted Karcher means and the weighted power means in strong operator
topology, as follows:

$\Lambda(\omega,\mathbb{A})=\lim_{t\rightarrow 0}P_{t}(\omega,\mathbb{A}).$
(3.3)

Moreover, they also showed that the following property holds for these means:

$P_{-t}(\omega;\mathbb{A})\leq\Lambda(\omega;\mathbb{A})\leq
P_{t}(\omega;\mathbb{A})\text{ for all }t\in(0,1].$ (3.4)

Let $\Phi$ be a unital positive linear map on $B(H)$. In [12] is proved that
if $t\in(0,1]$, then

$\Phi(P_{t}(\omega;\mathbb{A}))\leq P_{t}(\omega;\Phi(\mathbb{A})),$ (3.5)

where $\Phi(\mathbb{A})=(\Phi(A_{1}),\dots,\Phi(A_{n}))$.

In the following Theorem, we give a reverse inequality to the inequality
(3.5).

######  Theorem 3.

Let $\Phi$ be a unital positive linear map on $B(H)$ and $n\geq 2$ be a
positive integer. If $0<t\leq 1$ and $(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$.
Then

$\Phi(P_{t}(\omega;\mathbb{A}))\geq
K\left(h_{0},\frac{1}{2}\right)^{\frac{1}{t}}P_{t}(\omega;\Phi(\mathbb{A})),$
(3.6)

where $h_{0}=\max\limits_{1\leq i,j\leq n}R^{2}(A_{i},A_{j})$.

###### Proof.

Let $t\in(0,1]$ and $X_{t}=P_{t}(\omega;\mathbb{A})$. Then
$X_{t}=\sum_{i=1}^{n}\omega_{i}(X_{t}\sharp_{t}A_{i})$, by the operator
equation (3.2). From inequality (2.1), we have

$K\left(R^{2}(X_{t},A_{i}),t\right)(\Phi(X_{t})\sharp_{t}\Phi(A_{i}))\leq\Phi(X_{t}\sharp_{t}A_{i}).$
(3.7)

In the proof of proposition 3.5 of [11], was shown that

$d(P_{t}(\omega;\mathbb{A}),A_{j})\leq\max\limits_{1\leq i,j\leq
n}d(A_{i},A_{j}),$

Where $d$ is the Thompson metric. This implies that

$R^{2}(X_{t},A_{j})=R^{2}(P_{t}(\omega;\mathbb{A}),A_{j})\leq\max\limits_{1\leq
i,j\leq n}R^{2}(A_{i},A_{j})=h_{0}.$

Since the function
$K(h,\frac{1}{2})=\frac{2h^{\frac{1}{4}}}{1+h^{\frac{1}{2}}}$ is decreasing
for $h\geq 1$, Thus

$K\left(h_{0},\frac{1}{2}\right)\leq
K\left(R^{2}(X_{t},A_{i}),\frac{1}{2}\right)\leq
K\left(R^{2}(X_{t},A_{i}),t\right)$ (3.8)

Define $f(X)=\sum_{i=1}^{n}\omega_{i}(X\sharp_{t}\Phi(A_{i}))$. Then
$\lim_{n\rightarrow\infty}f^{n}(X)=P_{t}(\omega;\mathbb{A})$ for any $X>0$.
Let $\Phi$ be a unital positive linear map on $B(H)$. From relations (3.7) and
(3.8), we get

$\displaystyle\Phi(X_{t})$
$\displaystyle=\sum_{i=1}^{n}\omega_{i}\Phi(X_{t}\sharp_{t}A_{i})\geq\sum_{i=1}^{n}\omega_{i}K\left(R^{2}(X_{t},A_{i}),t\right)(\Phi(X_{t})\sharp_{t}\Phi(A_{i}))$
$\displaystyle\geq\sum_{i=1}^{n}\omega_{i}K\left(R^{2}(X_{t},A_{i}),\frac{1}{2}\right)(\Phi(X_{t})\sharp_{t}\Phi(A_{i}))$
$\displaystyle\geq\sum_{i=1}^{n}\omega_{i}K\left(h_{0},\frac{1}{2}\right)(\Phi(X_{t})\sharp_{t}\Phi(A_{i}))=K\left(h_{0},\frac{1}{2}\right)f(\Phi(X_{t})).$

Since $f$ is an increasing function, we have

$\displaystyle f\left(K\left(h_{0},\frac{1}{2}\right)^{-1}\Phi(X_{t})\right)$
$\displaystyle=\sum_{i=1}^{n}\omega_{i}\left(K\left(h_{0},\frac{1}{2}\right)^{-1}\Phi(X_{t})\sharp_{t}\Phi(A_{i})\right)$
$\displaystyle=K\left(h_{0},\frac{1}{2}\right)^{t-1}\sum_{i=1}^{n}\omega_{i}(\Phi(X_{t})\sharp_{t}\Phi(A_{i}))$
$\displaystyle=K\left(h_{0},\frac{1}{2}\right)^{t-1}f(\Phi(X_{t}))\geq
f^{2}(\Phi(X_{t})).$

Therefore

$\displaystyle\Phi(X_{t})\geq
K\left(h_{0},\frac{1}{2}\right)f(\Phi(X_{t}))\geq
K\left(h_{0},\frac{1}{2}\right)^{1+(1-t)}f^{2}(\Phi(X_{t})).$

Consequently

$\displaystyle\Phi(X_{t})\geq
K\left(h_{0},\frac{1}{2}\right)f(\Phi(X_{t}))\geq
K\left(h_{0},\frac{1}{2}\right)^{1+(1-t)+(1-t)^{2}+\dots+(1-t)^{n-1}}f^{n}(\Phi(X_{t})).$

This implies that

$\displaystyle\Phi(P_{t}(\omega;\mathbb{A}))=\Phi(X_{t})\geq
K\left(h_{0},\frac{1}{2}\right)^{\frac{1}{t}}\lim_{n\rightarrow\infty}f^{n}(\Phi(X_{t}))=K\left(h_{0},\frac{1}{2}\right)^{\frac{1}{t}}P_{t}(\omega;\Phi(\mathbb{A})).$

∎

Let $X$ be a positive definite operator on a Hilbert space $H$. Then
$m=\lambda_{min}(X)\leq X\leq\lambda_{max}(X)=M$, where
$\lambda_{min}(X)(\text{resp. }\lambda_{max}(X))$ is the minimum (resp.
maximum) of the specrum of $X$.

######  Theorem 4.

Let $\Phi$ be a unital positive linear map on $B(H)$ and $n\geq 2$ be a
positive integer. If $(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$. Then

$\Lambda(\omega;\Phi(\mathbb{A}))\geq\Phi(\Lambda(\omega;\mathbb{A}))\geq\frac{4\hbar}{(1+\hbar)^{2}}~{}\Lambda(\omega;\Phi(\mathbb{A})),$
(3.9)

where $m_{i}=\lambda_{min}(A_{i}),~{}M_{i}=\lambda_{max}(A_{i})$ and
$\hbar=\max\limits_{1\leq i\leq n}\frac{M_{i}}{m_{i}}$.

###### Proof.

From relations (3.3) and (3.5), we get

$\Phi(\Lambda(\omega;\mathbb{A}))\leq\Phi(P_{t}(\omega;\mathbb{A}))\leq
P_{t}(\omega;\Phi(\mathbb{A})).$ (3.10)

If $t\rightarrow 0$ in (3.10), then

$\Phi(\Lambda(\omega;\mathbb{A}))\leq\Lambda(\omega;\Phi(\mathbb{A})).$ (3.11)

Using (2.7), we obtain the following inequalities for
$\mathbb{A}=(A_{1},A_{2},\dots,A_{n})$

$\Phi(\mathbb{A})^{-1}\leq\Phi(\mathbb{A}^{-1})\leq
K(\hbar,2)\Phi(\mathbb{A})^{-1}=\Phi(K^{-1}(\hbar,2)\mathbb{A})^{-1}.$ (3.12)

Let $-1\leq t<0$, utilizing (3.5) and (3.12), we have

$\displaystyle\Phi(P_{t}(\omega;\mathbb{A}))$
$\displaystyle=\Phi\left(P_{-t}(\omega;\mathbb{A}^{-1}\right)^{-1})\geq\left(\Phi(P_{-t}(\omega;\mathbb{A}^{-1}))\right)^{-1}$
$\displaystyle\geq\left(P_{-t}(\omega;\Phi(\mathbb{A}^{-1}))\right)^{-1}\geq\left(P_{-t}(\omega;(\Phi(K^{-1}(\hbar,2)\mathbb{A)})^{-1})\right)^{-1}$
$\displaystyle\geq
P_{t}(\omega;\Phi(K^{-1}(\hbar,2)\mathbb{A)})=K^{-1}(\hbar,2)P_{t}(\omega;\Phi(\mathbb{A)}).$
(3.13)

From relation (3.3) and (3), we obtain

$K^{-1}(\hbar,2)P_{-t}(\omega;\Phi(\mathbb{A)})\leq\Phi(P_{-t}(\omega;\mathbb{A}))\leq\Phi(\Lambda(\omega;\mathbb{A}))\leq\Lambda(\omega;\Phi(\mathbb{A})).$
(3.14)

If $t\rightarrow 0$ in (3.14), then

$K^{-1}(\hbar,2)\Lambda(\omega;\Phi(\mathbb{A}))\leq\Phi(\Lambda(\omega;\mathbb{A}))\leq\Lambda(\omega;\Phi(\mathbb{A})).$

∎

##  4\. The geometric mean due to Ando-Li-Mathias

Arithmetic and harmonic means of $n$-operators can be defined, easily. But the
geometric mean case gives a lot of trouble because the product of operators is
non-commutative. Although several geometric means of $n$-operators are
defined, they do not have some important properties, for example, permutation
invariant or monotonicity. Some of mathematicians have long been interested to
extend the geometric mean of two operators to $n$-operators case. Ando, Lim
and Mathias in [2] suggested a good definition of the geometric mean for
extending it to raise the number of positive semi-defined matrices. It is
defined by a symmetric method and has many good properties. They listed ten
properties that a geometric mean of $m$ matrices should satisfy, and displayed
that their mean possesses all of them. Lawson and Lim in [10] have shown that
$G$ has all the ten properties. Other ideas of geometric mean with all the ten
properties have been suggested in [4, 6, 9]. In [20] Yamazaki pointed out that
definition of the geometric mean by Ando. Li and Mathias can be extended to
Hilbert space operators.

The geometric mean $G(A_{1},A_{2},\dots,A_{n})$ of any $n$-tuple of positive
definite operators $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$ is
defined by induction.

(i) $G(A_{1},A_{2})$ =$A_{1}\sharp A_{2}$

(ii) Assume that the geometric mean any $(n-1)$-tuple of operators is defined.
Let

$G((A_{j})_{j\neq i})=G(A_{1},A_{2},\dots,A_{i-1},A_{i+1},\dots,A_{n}),$

and let sequences $\\{\mathbb{A}_{i}^{(r)}\\}_{r=1}^{\infty}$ be
$\mathbb{A}_{i}^{(1)}=A_{i}$ and
$\mathbb{A}_{i}^{(r+1)}=G((\mathbb{A}_{j}^{(r)})_{j\neq i})$. If there exists
$\lim_{r\rightarrow\infty}{\mathbb{A}_{i}^{(r)}}$, and it does not depend on
$i$. Hence the geometric mean of $n$-operators is defined by

$\lim_{r\rightarrow\infty}{\mathbb{A}_{i}^{(r)}}=G((\mathbb{A}))=G(A_{1},A_{2},\dots,A_{n})\text{
for }i=1,\dots,n.$ (4.1)

In [2], Ando et al. showed there exists this limit and (4.1) is uniformly
convergence.

Furthermore, for $\mathbb{A}=(A_{1},\dots,A_{n})$ and
$\mathbb{B}=(B_{1},\dots,B_{n})\in\mathcal{P}^{n}$, the following important
inequality holds

$R(G(\mathbb{A}),G(\mathbb{B}))\leq\left(\prod_{i=1}^{n}R(A_{i},B_{i})\right)^{\frac{1}{n}}.$
(4.2)

Particular,

$R(\mathbb{A}_{i}^{(2)},\mathbb{A}_{k}^{(2)})=R(G((A_{j})_{j\neq
i}),G((A_{j})_{j\neq k}))\leq R(A_{i},A_{k})^{\frac{1}{n-1}}$ (4.3)

holds.

Yamazaki in [20] also obtained a converse of the arithmetic-geometric mean
inequality of $n$-operators via Kantorovich constant. Soon after, Fujii el al.
[8] also proved a stronger reverse inequality of the weighted arithmetic and
geometric means due to Lawson and Lim of $n$-operators by the Kantorovich
inequality.

Let $\mathbb{A}=(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$, $\Phi$ be a unital
positive linear map on $B(H)$. Then
$\Phi(A_{1}),\Phi(A_{2}),\dots,\Phi(A_{n})$ are $n$ positive definite
operators in $\mathcal{P}$. We consider the geometric mean
$G(\Phi(\mathbb{A}))$, where
$\Phi(\mathbb{A})=(\Phi(A_{1}),\Phi(A_{2}),\dots,\Phi(A_{n}))$, as follows:

$\displaystyle\Phi^{(1)}(\mathbb{A}_{i})$ $\displaystyle=\Phi(A_{i})\text{ for
}i=1,\dots,n,$

and for $r\geq 1,~{}i=1,\dots,n$

$\displaystyle\Phi^{(r+1)}(\mathbb{A}_{i})$
$\displaystyle=G((\Phi^{(r)}(\mathbb{A}_{i}))_{j\neq i})$
$\displaystyle=G(\Phi^{(r)}(\mathbb{A}_{1}),\dots,\Phi^{(r)}(\mathbb{A}_{i-1}),\Phi^{(r)}(\mathbb{A}_{i+1}),\dots,\Phi^{(r)}(\mathbb{A}_{n})).$

Finally,

$\displaystyle
G(\Phi(\mathbb{A}))=G((\Phi(A_{1}),\Phi(A_{2}),\dots,\Phi(A_{n})))=\lim_{r\rightarrow\infty}\Phi^{(r)}(\mathbb{A}_{i}).$

######  Theorem 5.

Let $n\geq 2$ be a positive integer, and
$(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$. Then

$\displaystyle\Phi(G(\mathbb{A}))\geq\left(\frac{2h_{1}^{\frac{1}{2}}}{1+h_{1}}\right)^{n-1}G(\Phi(\mathbb{A}))$
(4.4)

where $h_{1}=\max\limits_{1\leq i,j\leq n}R(A_{i},A_{j})$.

###### Proof.

First, for $r\geq 1$, we put $h_{r}=\max\limits_{1\leq i,j\leq
n}R(\mathbb{A}_{i}^{(r)},\mathbb{A}_{j}^{(r)})$ and
$K_{r}=K(h^{2}_{r},\frac{1}{2})$. By (4.3), we have

$1\leq h_{r}\leq h_{r-1}^{\frac{1}{n-1}}\leq\dots\leq
h_{1}^{(\frac{1}{n-1})^{r}}.$

Concavity the function $f(t)=t^{\alpha}$ for $0<\alpha\leq 1$ implies that

$\dfrac{1+t^{\alpha}}{2}\leq\left(\dfrac{1+t}{2}\right)^{\alpha},\text{ or
}\dfrac{2t^{\frac{\alpha}{2}}}{1+t^{\alpha}}\geq\left(\dfrac{2t^{\frac{1}{2}}}{1+t}\right)^{\alpha}.$

Since the function $K(t)=\dfrac{2t^{\frac{1}{2}}}{1+t}$ is a decreasing
function for $t\geq 1$, thus

$K_{r}=\dfrac{2h_{r}^{\frac{1}{2}}}{1+h_{r}}\geq\dfrac{2h_{1}^{\frac{1}{2}(\frac{1}{n-1})^{r}}}{1+h_{1}^{(\frac{1}{n-1})^{r}}}\geq\left(\dfrac{2h_{1}^{\frac{1}{2}}}{1+h_{1}}\right)^{(\frac{1}{n-1})^{r}}=K_{1}^{(\frac{1}{n-1})^{r}}.$
(4.5)

Now, we will prove (4.4) by induction on $n$. For $n=2$, from inequality
(2.5), we have

$\Phi(A_{1}\sharp A_{2})\geq K_{1}(\Phi(A_{1})\sharp\Phi(A_{2})).$

For $n=3$. A simple calculation shows that

$\displaystyle\Phi\Big{(}G(A_{1}\sharp A_{3},A_{1}\sharp A_{2})\Big{)}$
$\displaystyle=\Phi\Big{(}(A_{1}\sharp A_{3})\sharp(A_{1}\sharp
A_{2})\Big{)}\geq K_{2}\Big{[}\Phi\Big{(}A_{1}\sharp
A_{3}\Big{)}\sharp\Phi\Big{(}A_{1}\sharp A_{2}\Big{)}\Big{]}$
$\displaystyle\geq
K_{2}\Big{[}K_{1}\Big{(}\Phi(A_{1})\sharp\Phi(A_{3})\Big{)}\sharp
K_{1}\Big{(}\Phi(A_{1})\sharp\Phi(A_{2})\Big{)}\Big{]}$
$\displaystyle=K_{2}K_{1}\Big{[}\Big{(}\Phi(A_{1})\sharp\Phi(A_{3})\Big{)}\sharp\Big{(}\Phi(A_{1})\sharp(\Phi(A_{2})\Big{)}\Big{]}.$
$\displaystyle=K_{2}K_{1}G\Big{(}\Phi(A_{1})\sharp\Phi(A_{3}),\Phi(A_{1})\sharp\Phi(A_{2})\Big{)}.$

Therefore $\Phi(A_{1}^{(3)})\geq K_{2}K_{1}\Phi^{(3)}(A_{1})$. Hence for
$r\geq 1$, we obtain

$\Phi(A_{i}^{(r)})\geq(K_{r-1}K_{r-2}\dots K_{1})\Phi^{(r)}(A_{i}),\text{ for
}i=1,2,3.$

Assume that Theorem 2 holds for $n-1$. We prove the case $n$. From the
induction hypothesis, we have

$\displaystyle\Phi(A_{i}^{(r)})$ $\displaystyle=\Phi(G((A_{j}^{(r-1)})_{j\neq
i}))=\Phi\left(G\big{(}A_{1}^{(r-1)},A_{2}^{(r-1)},\dots,A_{i+1}^{(r-1)},\dots,A_{n}^{(r-1)}\big{)}\right)$
$\displaystyle\geq
K_{r-1}^{n-2}G\big{(}\Phi(A_{1}^{(r-1)}),\dots,\Phi(A_{i+1}^{(r-1)}),\dots,\Phi(A_{n}^{(r-1)})\big{)}$
$\displaystyle\geq
K_{r-1}^{n-2}G\big{(}K_{r-2}^{n-2}\Phi(A_{1}^{(r-1)}),\dots,K_{r-2}^{n-2}\Phi(A_{i+1}^{(r-1)}),\dots,K_{r-2}^{n-2}\Phi(A_{n}^{(r-1)})\big{)}$
$\displaystyle\geq
K_{r-1}^{n-2}K_{r-2}^{n-2}G\big{(}\Phi(A_{1}^{(r-1)}),\dots,\Phi(A_{i+1}^{(r-1)}),\dots,\Phi(A_{n}^{(r-1)})\big{)}$
$\displaystyle=K_{r-1}^{n-2}K_{r-2}^{n-2}G\big{(}\Phi(G(A_{1}^{(r-2)}),\dots,\Phi(G(A_{i+1}^{(r-2)}),\dots,\Phi(G(A_{n}^{(r-2)})\big{)}$
$\displaystyle\vdots$ $\displaystyle\geq(K_{r-1}K_{r-2}\dots
K_{1})^{n-2}G\big{(}\Phi(A_{1}),\dots,\Phi(A_{i+1}),\dots,\Phi(A_{n})\big{)}$
$\displaystyle=(K_{r-1}K_{r-2}\dots K_{1})^{n-2}G(\Phi(A_{j})_{j\neq i}),$

therefore, for positive integer $i=1,2,\dots,n$, we deduce

$\Phi(A_{i}^{(r)})\geq(K_{r-1}K_{r-2}\dots K_{1})^{n-2}G(\Phi(A_{j})_{j\neq
i}).$ (4.6)

From (4.5), we get

$K_{r-1}K_{r-2}\dots K_{1}\geq K_{1}K_{1}^{\frac{1}{n-1}}\dots
K_{1}^{(\frac{1}{n-1})^{r-1}}=K_{1}^{1+\frac{1}{n-1}+\dots+(\frac{1}{n-1})^{r-1}}.$

Consequently

$\liminf_{r\rightarrow\infty}K_{r-1}K_{r-2}\dots,K_{1}\geq
K_{1}^{\frac{n-1}{n-2}},$ (4.7)

since $K_{1}^{1+\frac{1}{n-1}+\dots+(\frac{1}{n-1})^{r-1}}\rightarrow
K_{1}^{\frac{n-1}{n-2}}$. We know that

$\lim_{r\rightarrow\infty}\Phi(A_{i}^{(r)})=\Phi(\lim_{r\rightarrow\infty}A_{i}^{(r)})=\Phi(G(A_{1},A_{2},\dots,A_{n}))$
(4.8)

and

$\displaystyle\lim_{r\rightarrow\infty}\Phi^{(r)}(A_{i})=G(\Phi(A_{1}),\Phi(A_{2}),\dots,\Phi(A_{n})).$
(4.9)

From inequalities (4.6), (4.7), (4.8) and (4.9), we obtain the desired
inequality (4.4).

∎

######  Corollary 1.

Let $n\geq 2$ be a positive integer and $(A_{1},\dots,A_{n})$ be a $n$-tuple
in $\mathcal{P}^{n}$. If $0<m_{i}I\leq A_{i}\leq M_{i}I$ for some scalers
$0<m_{i}<M_{i}~{}(i=1,2,\dots,n)$. Then

$\left(\frac{2\sqrt{M_{0}}}{1+M_{0}}\right)^{n-1}\left(\prod_{j=1}^{n}\langle
A_{j}x,x\rangle\right)^{\frac{1}{n}}\leq\langle
G(A_{1},\dots,A_{n})x,x\rangle\leq\left(\prod_{j=1}^{n}\langle
A_{j}x,x\rangle\right)^{\frac{1}{n}},$ (4.10)

where $M_{0}=\max\limits_{1\leq i,j\leq n}\\{\frac{M_{i}}{m_{j}}\\}$.

###### Proof.

Let $(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$. The relation $0<m_{i}I\leq
A_{i}\leq M_{i}I$ implies that

$\displaystyle
R(A_{i},A_{j})=\max\\{r(A_{i}^{-1}A_{j}),r(A_{j}^{-1}A_{i})\\}\leq\max\left\\{\frac{M_{j}}{m_{i}},\frac{M_{i}}{m_{j}}\right\\}.$

Therefore

$h_{0}=\max_{1\leq i,j\leq n}R(A_{i},A_{j})\leq\max_{1\leq i,j\leq
n}\left\\{\frac{M_{i}}{m_{j}}\right\\}=M_{0}.$

This implies that

$K(h^{2}_{0},\frac{1}{2})\geq
K(M^{2}_{0},\frac{1}{2})=\frac{2\sqrt{M_{0}}}{1+M_{0}}.$ (4.11)

It can easily be verified by the iteration argument from the two variable case
that the following inequality holds

$\Phi(G(A_{1},A_{2},\dots,A_{n}))\leq
G(\Phi(A_{1}),\Phi(A_{2}),\dots,\Phi(A_{n})).$ (4.12)

Applying (4.4) and (4.12) to the positive linear functional $\Phi(A)=\langle
Ax,x\rangle$, where $x$ is a unit vector in $H$, we get the inequalities in
(4.10) with Kantorovich constant $K(h^{2}_{0},\frac{1}{2})$. The desired
inequalities in (4.10) is obtained by (4.11). ∎

In particular, if $0<mI\leq A_{i}\leq MI$, then $M_{0}=\frac{M}{m}$ and the
relation (4.10) becomes

$\left(\frac{2\sqrt{mM}}{m+M}\right)^{n-1}\left(\prod_{j=1}^{n}\langle
A_{j}x,x\rangle\right)^{\frac{1}{n}}\leq\langle
G(A_{1},\dots,A_{n})x,x\rangle\leq\left(\prod_{j=1}^{n}\langle
A_{j}x,x\rangle\right)^{\frac{1}{n}}.$

From inequalities (4.4) for $\Phi(A)=\langle Ax,x\rangle$ and as regards for
positive linear operators $\|A\|=\sup\\{\langle Ax,x\rangle:~{}\|x\|\leq
1\\}$, we give the following result:

######  Corollary 2.

Let $n\geq 2$ be a positive integer, $(A_{1},\dots,A_{n})\in\mathcal{P}^{n}$
and let $0<m_{i}I\leq A_{i}\leq M_{i}I$ for $i=1,2,\dots,n$ and for some
scalers $0<m_{i}<M_{i}$. Then

$\left(\frac{2\sqrt{M_{0}}}{1+M_{0}}\right)^{n-1}\prod_{j=1}^{n}\|A_{j}\|^{\frac{1}{n}}\leq\|G(A_{1},\dots,A_{n})\|\leq\prod_{j=1}^{n}\|A_{j}\|^{\frac{1}{n}},$
(4.13)

where $M_{0}=\max\limits_{1\leq i,j\leq n}\\{\frac{M_{i}}{m_{j}}\\}$.

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