# Tail bounds for Multivariate Random Tensor Means

Shih Yu Chang Shih Yu Chang is with the Department of Applied Data Science,
San Jose State University, San Jose, CA, U. S. A. (e-mail:
shihyu.chang@sjsu.edu).

###### Abstract

In our recent research endeavors, we have delved into the realm of tail bounds
problems concerning bivariate random tensor means. In this context, tensors
are treated as finite-dimensional operators. However, the longstanding
challenge of extending the concept of operator means to scenarios involving
more than two variables had persisted. The primary objective of this present
study is to unveil a collection of tail bounds applicable to multivariate
random tensor means. These encompass the weighted arithmetic mean, weighted
harmonic mean, and the Karcher mean. These bounds are derived through the
utilization of Ando-Hiai’s inequalities, alongside tail bounds specifically
tailored for multivariate random tensor means employing reverse Ando-Hiai’s
inequalities, which are rooted in Kantorovich constants. Notably, our
methodology involves employing the concept of deformation for operator means
with multiple variables, following the principles articulated in Hiai, Seo and
Wada’s recent work. Additionally, our research contributes to the expansion of
the Karcher mean differentiable region from the vicinity of the diagonal
identity element within the Cartesian product space of positive definite
tensors to the vicinity of the general element within the Cartesian product
space of positive definite tensors via the application of the inverse and
implicit function theorem.

Index terms— Löwner ordering, Ando-Hiai inequality, random tensors, geometric
mean, Karcher mean, power mean, Kantorovich constant

##  1 Introduction

Tensors are used in science and technologies for several reasons due to their
versatility and ability to represent complex relationships and data
structures. We discuss several important examples here. First, tensors are
used to represent multidimensional data. Many real-world problems involve data
that have multiple dimensions, such as time-series data, images, audio
signals, and more. Tensors provide a natural and efficient way to represent
and manipulate such data, making them essential in fields like image
processing, computer vision, speech recognition, and signal processing.
Second, tensors are fundamental building blocks in deep learning and neural
networks. Neural networks process data in the form of tensors, and tensor-
based operations, like matrix multiplications, are at the core of training and
inference processes. This has led to remarkable advancements in various fields
like computer vision, natural language processing, and robotics. Third, in
physics and engineering, many systems are described by equations involving
tensors. For example, in fluid dynamics, stress tensors describe the behavior
of fluids under different conditions. Tensors are also used in
electromagnetism, solid mechanics, and general relativity, among other
disciplines, to represent physical quantities and their interactions. Fourth,
in addition to deep learning, tensors are used in other machine learning
techniques, like tensor decomposition methods and tensor-based data analysis.
These methods are employed for data compression, dimensionality reduction, and
feature extraction. Finally, tensors play a crucial role in robotics for
representing spatial transformations and manipulator kinematics. They are used
in control systems to model and control complex multi-input and multi-output
systems. Overall, tensors offer a powerful and unified framework to handle
multi-dimensional data and complex relationships, making them indispensable in
various scientific and technological domains. Their ability to capture and
process intricate information allows researchers from science and technology
fields to solve challenging problems and make significant advancements in
their respective fields [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

In our recent works, we consider tail bounds problems for bivariate random
tensor means, where tensors can be treated as finite dimensional operators
[13, 14]. However, the problem of extending the operator mean to more than two
variables of matrices or operators had been a longstanding challenge.It was
eventually resolved through two different approaches: an iteration method [15]
and a Riemannian geometry method [16]. Subsequently, the Riemannian geometry
method has been significantly advanced by various authors, including [17]. In
the present day, the multivariate geometric mean within the Riemannian
geometric approach is commonly referred to as the _Karcher mean_ as it is
determined by solving the so-called “Karcher equation”. Additionally, another
essential multivariate operator mean that has garnered recent active attention
is the power mean, which was developed in [18]. The purpose of this work is to
present several tail bounds for multivariate random tensor means, e.g.,
weighted arithmetic mean, weighted harmonic mean, Karcher mean, based on Ando-
Hiai’s inequalities and tail bounds for multivariate random tensor means with
reverse Ando-Hiai’s inequalities derived from Kantorovich constants. The
technique of deformation for $k$-variable operator means is used here
according to [19, 20]. Besides, we also extend Theorem 6.3 in [18] about the
Karcher mean differentiable region from the neighbor of the diagonal identity
element in the Cartesian product space of positive definite tensors to the
neighbor of general elements in the Cartesian product space of positive
definite tensors by applying the inverse and implicit function theorem.

The rest of this paper is organized as follows. In Section 2, we will review
basic definitions about tensors used in this work, especially the notion about
RTT. In Section 3, multivariate random tensor means and multivariate random
tensor deformed means, and their associated examples like weighted arithmetic
mean, weighted harmonic mean, Karcher mean, will be discussed. The continuity
and differentiability of the Karcher mean will be discussed in Section 4. In
Section 5, we will derive tail bounds for multivariate random tensor means. In
Section 6, we will generalize tail bounds from multivariate random tensor mean
to multivariate random tensor deformed means. Finally, in Section 7, we
consider the reverse of Ando-Hiai Type inequalities via Kantorovich constants
for multivariate random tensor means and multivariate random tensor deformed
means.

Nomenclature: The sets of complex and real numbers are denoted by $\mathbb{C}$
and $\mathbb{R}$, respectively. The set of natural numbers is represented by
$\mathbb{N}$. A scalar is denoted by an either italicized or Greek alphabet
such as $x$ or $\beta$; a vector is denoted by a lowercase bold-faced alphabet
such as $\bm{x}$; a matrix is denoted by an uppercase bold-faced alphabet such
as $\bm{X}$; a tensor is denoted by a calligraphic alphabet such as
$\mathcal{x}$ or $\mathcal{X}$.

##  2 Tensor Preliminaries

In this section, we will review required definitions and basic facts about
tensors. The default product between two tensors
$\mathcal{A}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ and
$\mathcal{B}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ is Einstein product with order $N$, denoted by
$\star_{N}$. We will specfiy the exact product symbol if the product is not
$\star_{N}$.

As the matrix eigen-decomposition theorem is crucial in various linear algebra
theory and applications, we will have a parallel decomposition theorem for
Hermitian tensors. From Theorem 5.2 in [21], every Hermitian tensor
$\mathcal{H}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ has the following decomposition:

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{r}\lambda_{i}\mathcal{U}_{i}\star_{1}\mathcal{U}^{H}_{i},\mbox{
\leavevmode\nobreak\ with\leavevmode\nobreak\ \leavevmode\nobreak\
$\langle\mathcal{U}_{i},\mathcal{U}_{i}\rangle=1$ and
$\langle\mathcal{U}_{i},\mathcal{U}_{j}\rangle=0$ for $i\neq j$,}$ (1)

where $\lambda_{i}\in\mathbb{R}$ and
$\mathcal{U}_{i}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times 1}$. Here
tensors $\mathcal{U}_{i}$ are orthogonal tensors each other since
$\langle\mathcal{U}_{i},\mathcal{U}_{j}\rangle=0$ for $i\neq j$. The values
$\lambda_{i}$ are named as _Hermitian eigenvalues_ , and the minimum integer
of $r$ to decompose a Hermitian tensor as in Eq. (1) is called _Hermitian
tensor rank_. In this work, we assume that all Hermitian tensors discussed in
this work are full rank, i.e., $r=\prod\limits_{j=1}^{N}I_{j}$. A _positive
definite_ (PD) tensor is a Hermitian tensor with all _Hermitian eigenvalues_
are positive. A _semipositive definite_ (SPD) tensor is a Hermitian tensor of
which all _Hermitian eigenvalues_ are nonnegative. The symbols
$\lambda_{\min}(\mathcal{X})$ and $\lambda_{\max}(\mathcal{X})$ will represent
the minimum and maximum eigenvalues of the tensor $\mathcal{X}$. Given two
Hermitian tensors $\mathcal{X}$ and $\mathcal{Y}$, we write
$\mathcal{X}\preceq\mathcal{Y}$ if $\mathcal{Y}-\mathcal{X}$ is a SPD tensor,
and write $\mathcal{X}\succeq\mathcal{Y}$ if $\mathcal{X}-\mathcal{Y}$ is a
SPD tensor. We adopt $S_{\mbox{\tiny PD}}$ to represent a set of PD tensors,
and $S_{\mbox{\tiny PSD}}$ to represent a set of SPD tensors.

Let us represent the Hermitian eigenvalues of a Hermitian tensor
$\mathcal{H}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ in decreasing order by the vector
$\vec{\lambda}(\mathcal{H})=(\lambda_{1}(\mathcal{H}),\cdots,\lambda_{r}(\mathcal{H}))$,
where $r$ is the Hermitian rank of the tensor $\mathcal{H}$. We use
$\mathbb{R}_{\geq 0}(\mathbb{R}_{>0})$ to represent a set of nonnegative
(positive) real numbers. Let $\left\|\cdot\right\|_{\rho}$ be a unitarily
invariant tensor norm, i.e.,
$\left\|\mathcal{H}\star_{N}\mathcal{U}\right\|_{\rho}=\left\|\mathcal{U}\star_{N}\mathcal{H}\right\|_{\rho}=\left\|\mathcal{H}\right\|_{\rho}$,
where $\mathcal{U}$ is for unitary tensor. Let $\rho:\mathbb{R}_{\geq
0}^{r}\rightarrow\mathbb{R}_{\geq 0}$ be the corresponding gauge function that
satisfies H${\rm\ddot{o}}$lder’s inequality so that

$\displaystyle\left\|\mathcal{H}\right\|_{\rho}=\left\||\mathcal{H}|\right\|_{\rho}=\rho(\vec{\lambda}(|\mathcal{H}|)),$
(2)

where $|\mathcal{H}|\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sqrt{\mathcal{H}^{H}\star_{N}\mathcal{H}}$.

The notion about _random tensor topology_ (RTT) is provided by the following
definition.

###### Definition 1

We say that a sequence of random tensor
$\mathcal{X}_{n}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ converges to the random tensor
$\mathcal{X}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ with respect to the tensor norm
$\left\|\cdot\right\|_{\rho}$ in the sense of RTT, if we have

$\displaystyle\mathbb{E}\left(\left\|\mathcal{X}_{n}\right\|_{\rho}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{exists,}$ (3)

and

$\displaystyle\lim\limits_{n\rightarrow\infty}\mathbb{E}\left(\left\|\mathcal{X}_{n}-\mathcal{X}\right\|_{\rho}\right)=0.$
(4)

We adopt the notation
$\lim\limits_{n\rightarrow\infty}\mathcal{X}_{n}=\mathcal{X}$ to represent
that random tensors $\mathcal{X}_{n}$ converges to the random tensor
$\mathcal{X}$ with respect to the tensor norm $\left\|\cdot\right\|_{\rho}$ in
the sense of RTT.

##  3 Multivariate Random Tensor Means

###  3.1 Multivariate Random Tensor Means and Deformed Means

We have to introduce the Thompson metric first. The Thompson metric between
two PD tensors $\mathcal{X},\mathcal{Y}\in\mathbb{C}^{I_{1}\times\dots\times
I_{N}\times I_{1}\times\dots\times I_{N}}$, represented by
$d_{T}\left(\mathcal{X},\mathcal{Y}\right)$ on PD tensors is defined by

$\displaystyle
d_{T}\left(\mathcal{X},\mathcal{Y}\right)\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\lambda_{\max}\left(\log\left(\mathcal{X}^{-1/2}\star_{N}\mathcal{Y}\star_{N}\mathcal{X}^{-1/2}\right)\right)=\log\max\\{\bm{\alpha}(\mathcal{X}/\mathcal{Y}),\bm{\alpha}(\mathcal{Y}/\mathcal{X})\\},$
(5)

where $\bm{\alpha}(\mathcal{X}/\mathcal{Y})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\inf\\{\alpha>0:\mathcal{X}\preceq\alpha\mathcal{Y}\\}$ [22].

The concept of bivariate operator means was formulated as an axiomatic way by
Kubo-Ando [23]. Let $\sigma:S_{\mbox{\tiny PD}}\times S_{\mbox{\tiny
PD}}\rightarrow S_{\mbox{\tiny PD}}$ be a map of _bivaraite tensor mean_ , and
all random tensors used in the following list with dimensions
$\mathbb{C}^{I_{1}\times\dots\times I_{N}\times I_{1}\times\dots\times
I_{N}}$, we extend this notion to random tensors as follows:

  1. (i)

Monotonicity: $\mathcal{X}\preceq\mathcal{Z}$, $\mathcal{Y}\preceq\mathcal{W}$
$\Longrightarrow$
$\mathcal{X}\sigma\mathcal{Y}\preceq\mathcal{Z}\sigma\mathcal{W}$,  almost
surely;

  2. (ii)

Transformer Inequality:

$\displaystyle\mathcal{Z}\star_{N}(\mathcal{X}\sigma\mathcal{Y})\star_{N}\mathcal{Z}\preceq(\mathcal{Z}\star_{N}\mathcal{X}\star_{N}\mathcal{Z})\sigma(\mathcal{Z}\star_{N}\mathcal{Y}\star_{N}\mathcal{Z}),\mbox{\leavevmode\nobreak\
almost surely},$ (6)

where $\mathcal{Z}\in S_{\mbox{\tiny SPD}}$;

  3. (iii)

Monotone continuity: $\mathcal{X}_{n}\searrow\mathcal{X}$,
$\mathcal{Y}_{n}\searrow\mathcal{Y}$, $\Longrightarrow$
$\mathcal{X}_{n}\sigma\mathcal{Y}_{n}\searrow\mathcal{X}\sigma\mathcal{Y}$,
where $\mathcal{X}_{n}\searrow\mathcal{X}$ indicates that
$\mathcal{X}_{1}\succeq\mathcal{X}_{2}\succeq\cdots$ and
$\lim\limits_{n\rightarrow\infty}\mathcal{X}_{n}=\mathcal{X}$ in RTT, on the
other hand, $\mathcal{X}_{n}\nearrow\mathcal{X}$,
$\mathcal{Y}_{n}\nearrow\mathcal{Y}$, $\Longrightarrow$
$\mathcal{X}_{n}\sigma\mathcal{Y}_{n}\nearrow\mathcal{X}\sigma\mathcal{Y}$,
where $\mathcal{X}_{n}\nearrow\mathcal{X}$ indicates that
$\mathcal{X}_{1}\preceq\mathcal{X}_{2}\preceq\cdots$ and
$\lim\limits_{n\rightarrow\infty}\mathcal{X}_{n}=\mathcal{X}$ in RTT;

  4. (iv)

Normalized condition: $\mathcal{I}\sigma\mathcal{I}=\mathcal{I}$.

Note that transformer inequality will become equal if the tensor $\mathcal{Z}$
belongs to $S_{\mbox{\tiny PD}}$ (invertible).

In [23], they showed that there is a one-to-one correspondence between the
operator mean $\sigma$ and the non-negative operator monotone function
$g_{\sigma}$ on $(0,\infty)$ with $g(1)=1$ determined by

$\displaystyle g(x)\mathcal{I}$ $\displaystyle=$
$\displaystyle\mathcal{I}\sigma(x\mathcal{I}),$ (7)

where $x>0$; and

$\displaystyle\mathcal{X}\sigma\mathcal{Y}$ $\displaystyle=$
$\displaystyle\mathcal{X}^{-1/2}\star_{N}g(\mathcal{X}^{1/2}\star_{N}\mathcal{Y}\star_{N}\mathcal{X}^{-1/2})\star_{N}\mathcal{X}^{1/2},$
(8)

where $\mathcal{X},\mathcal{Y}\in\mathbb{C}^{I_{1}\times\dots\times
I_{N}\times I_{1}\times\dots\times I_{N}}$ are PD tensors. Such corresponding
function $g_{\sigma}$ is named as _corresponding function_ of $\sigma$. We say
that the operator $\sigma$ is _power monotone increasing_ (p.m.i.) if
$g_{\sigma}(x^{p})\geq g^{p}_{\sigma}(x)$ for all $x>0$ and $p\geq 1$. Using
the limitation method given by [14], we can extend Eq. (8) from PD tensors to
SPD tensors in RTT.

The bivariate Kubo-Ando axioms provided in Section (3.1) can be extended to
the multivariate version given in Section (3.1). Let
$\mathfrak{M}:S^{k}_{\mbox{\tiny PD}}\rightarrow S_{\mbox{\tiny PD}}$ be a map
of _$k$ -variable tensor mean_, and all random tensors used in the following
list with dimensions $\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$, the multivariate Kubo-Ando axioms is given by
the follows:

  1. (I)

Monotonicity: If $\mathcal{X}_{i},\mathcal{Y}_{i}\in S_{\mbox{\tiny PD}}$ and
$\mathcal{X}_{i}\preceq\mathcal{Y}_{i}$ for $1\leq i\leq k$, then we have
$\mathfrak{M}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\preceq\mathfrak{M}(\mathcal{Y}_{1},\cdots,\mathcal{Y}_{k})$,
almost surely;

  2. (II)

Congruence invariance: For every $\mathcal{X}_{i}\in S_{\mbox{\tiny PD}}$,
where $1\leq i\leq k$ and any invertible tensor $\mathcal{A}$, we have

$\displaystyle\mathcal{A}^{\mathrm{H}}\star_{N}\mathfrak{M}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\star_{N}\mathcal{A}$
$\displaystyle=$
$\displaystyle\mathfrak{M}(\mathcal{A}^{\mathrm{H}}\star_{N}\mathcal{X}_{1}\star_{N}\mathcal{H},\cdots,\mathcal{A}^{\mathrm{H}}\star_{N}\mathcal{X}_{k}\star_{N}\mathcal{H}),$
(9)

where $\mathrm{H}$ is a Hermitan transpose;

  3. (III)

Monotone continuity: $\mathcal{X}_{i},\mathcal{X}_{i,n}\in S^{k}_{\mbox{\tiny
PD}}$ for $1\leq i\leq k$ and $n\in\mathbb{N}$. If either
$\mathcal{X}_{i,n}\searrow\mathcal{X}_{i}$, or
$\mathcal{X}_{i,n}\nearrow\mathcal{X}_{i}$ as $n\rightarrow\infty$ for each
$i$, we have

$\displaystyle\lim\limits_{n\rightarrow\infty}\mathfrak{M}(\mathcal{X}_{1,n},\cdots,\mathcal{X}_{k,n})$
$\displaystyle=$
$\displaystyle\mathfrak{M}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$ (10)

where the limitation is in the sense of RTT;

  4. (IV)

Normalized condition:
$\mathfrak{M}(\mathcal{I},\cdots,\mathcal{I})=\mathcal{I}$.

Given $k$ PD tensors, $\mathcal{A}_{1},\cdots,\mathcal{A}_{k}$, we consider
the following tensor-valued equation

$\displaystyle\mathcal{X}$ $\displaystyle=$
$\displaystyle\mathfrak{M}(\mathcal{X}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}\sigma\mathcal{A}_{k}).$
(11)

The solution $\mathcal{X}$ of Eq. (11) shall play an important role in
determining the multivariate tensor mean with respect
$\mathcal{A}_{1},\cdots,\mathcal{A}_{k}$.

Before presenting the main result of this section, we need following lemmas
about the Thompson metric.

###### Lemma 1

For every $\mathcal{X}_{i},\mathcal{Y}_{i}\in S_{\mbox{\tiny PD}}$, where
$1\leq i\leq k$, we have

$\displaystyle
d_{T}(\mathfrak{M}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),\mathfrak{M}(\mathcal{Y}_{1},\cdots,\mathcal{Y}_{k}))\leq\max\limits_{1\leq
i\leq
k}d_{T}(\mathfrak{M}(\mathcal{X}_{i},\mathcal{Y}_{i}),\leavevmode\nobreak\
\mbox{almost surely.}$ (12)

Proof: Let $\kappa>0$ be the largest term for
$d_{T}(\mathfrak{M}(\mathcal{X}_{i},\mathcal{Y}_{i})$ among all $i$, we have

$\displaystyle e^{-\kappa}\mathcal{X}_{i}\preceq\mathcal{Y}_{i}\preceq
e^{\kappa}\mathcal{X}_{i},\leavevmode\nobreak\ \mbox{almost surely.}$ (13)

Then, from axioms (I) and (II) in the multivariate Kubo-Ando axioms given at
Section (3.1), we have

$\displaystyle\mathfrak{M}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})$
$\displaystyle=$
$\displaystyle\mathfrak{M}(e^{-\kappa}\mathcal{X}_{1},\cdots,e^{-\kappa}\mathcal{X}_{k})$
(14) $\displaystyle\preceq$
$\displaystyle\mathfrak{M}(\mathcal{Y}_{1},\cdots,\mathcal{Y}_{k}),\leavevmode\nobreak\
\mbox{almost surely;}$

and

$\displaystyle\mathfrak{M}(\mathcal{Y}_{1},\cdots,\mathcal{Y}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{M}(e^{\kappa}\mathcal{X}_{1},\cdots,e^{\kappa}\mathcal{X}_{k})$
(15) $\displaystyle=$ $\displaystyle
e^{\kappa}\mathfrak{M}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),\leavevmode\nobreak\
\mbox{almost surely.}$

Then, this Lemma is proved from Eqs. (14) and (15). $\hfill\Box$

The next Lemma shows that the Thompson metric will be reduced by taking the
mean with respect to a PD tensor.

###### Lemma 2

Given $\mathcal{X},\mathcal{Y},\mathcal{A}\in S_{\mbox{\tiny PD}}$ and
$\mathcal{X}\neq\mathcal{Y}$, then
$d_{T}(\mathcal{X}\sigma\mathcal{A},\mathcal{Y}\sigma\mathcal{A})<d_{T}(\mathcal{X},\mathcal{Y})$.

Proof: We wish to show that if $\mathcal{A}\prec\mathcal{B}$ is almost surely,
we have $\mathcal{X}\sigma\mathcal{A}\prec\mathcal{X}\sigma\mathcal{B}$ is
almost surely. Let $g_{\sigma}$ be the one-to-one correspondence function with
respect to the tensor mean $\sigma$, then we have

$\displaystyle\mathcal{X}\sigma\mathcal{A}$ $\displaystyle=$
$\displaystyle\mathcal{X}^{1/2}g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}\mathcal{X}^{-1/2})\mathcal{X}^{1/2}$
(16) $\displaystyle\prec$
$\displaystyle\mathcal{X}^{1/2}g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{B}\mathcal{X}^{-1/2})\mathcal{X}^{1/2}$
$\displaystyle=$ $\displaystyle\mathcal{X}\sigma\mathcal{B},$

where we use that the function $g_{\sigma}$ is strictly tensor increasing on
$(0,\infty)$.

Set $\kappa\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}d_{T}(\mathcal{X},\mathcal{Y})>0$ and the fact that
$e^{-\kappa}\mathcal{A}\prec\mathcal{A}\prec e^{\kappa}\mathcal{A}$ for any
tensor $\mathcal{A}$, we have

$\displaystyle\mathcal{Y}\sigma\mathcal{A}\preceq
e^{\kappa}\mathcal{X}\sigma\mathcal{A}\prec e^{\kappa}\mathcal{X}\sigma
e^{\kappa}\mathcal{A}=e^{\kappa}(\mathcal{X}\sigma\mathcal{A}),$ (17)

and

$\displaystyle\mathcal{Y}\sigma\mathcal{A}\succeq
e^{-\kappa}\mathcal{X}\sigma\mathcal{A}\succ e^{-\kappa}\mathcal{X}\sigma
e^{-\kappa}\mathcal{A}=e^{-\kappa}(\mathcal{X}\sigma\mathcal{A}).$ (18)

From Eqs. (17), (18) and Thompson meetric definition provided by Eq. (5), this
Lemma is proved. $\hfill\Box$

We are ready to present the main result of this section.

###### Theorem 1

  1. (I)

Given $k$ PD tensors, $\mathcal{A}_{1},\cdots,\mathcal{A}_{k}$, there exists a
unique solution $\mathcal{X}\in S_{\mbox{\tiny PD}}$ satisfying the following:

$\displaystyle\mathcal{X}$ $\displaystyle=$
$\displaystyle\mathfrak{M}(\mathcal{X}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}\sigma\mathcal{A}_{k});$
(19)

  2. (II)

Let $\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$ be the
unique solution of Eq. (19), then $\mathfrak{M}_{\sigma}:S^{k}_{\mbox{\tiny
PD}}\rightarrow S_{\mbox{\tiny PD}}$ is a $k$-variable mean satisfying Kubo-
Ando axioms given at Section (3.1);

  3. (III)

If $\mathcal{X}\in S_{\mbox{\tiny PD}}$ and
$\mathcal{X}\preceq\mathfrak{M}(\mathcal{X}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}\sigma\mathcal{A}_{k})$,
then
$\mathcal{X}\preceq\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$;

  4. (IV)

If $\mathcal{X}\in S_{\mbox{\tiny PD}}$ and
$\mathcal{X}\succeq\mathfrak{M}(\mathcal{X}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}\sigma\mathcal{A}_{k})$,
then
$\mathcal{X}\succeq\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.

Proof: For item (I) proof, we first select a constant $\alpha>0$ such that
$\mathcal{A}_{1},\cdots,\mathcal{A}_{k}\in S_{\alpha}$, where $S_{\alpha}$ is
a set defined by $S_{\alpha}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{\mathcal{X}\in S_{\mbox{\tiny
PD}}:\alpha^{-1}\mathcal{I}\preceq\mathcal{X}\preceq\alpha\mathcal{I}\\}$. We
also define the map $G:S_{\mbox{\tiny PD}}\rightarrow S_{\mbox{\tiny PD}}$ by

$\displaystyle G(\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$
(20)

From the multivariate Kubo-Ando axioms given at Section (3.1), we have
$G(\mathcal{X})\in S_{\alpha}$ if $\mathcal{X}\in S_{\alpha}$ and monotonicity
of $G(\mathcal{X})$. If we set
$\mathcal{Y}_{0}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\alpha\mathcal{I}$, we have $\mathcal{Y}_{0}\succeq
G(\mathcal{Y}_{0})\succeq
G^{2}(\mathcal{Y}_{0})\succeq\cdots\succeq\alpha^{-1}\mathcal{I}$, from (III)
at the multivariate Kubo-Ando axioms given at Section (3.1), we have
$G^{n}(\mathcal{Y}_{0})\searrow\mathcal{X}_{0}$ for some $\mathcal{X}_{0}\in
S_{\mbox{\tiny PD}}$. Then, we have

$\displaystyle\mathcal{X}_{0}$ $\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}G(G^{n}(\mathcal{Y}_{0}))$ (21)
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\mathfrak{M}(G^{n}(\mathcal{Y}_{0})\sigma\mathcal{A}_{1},\cdots,G^{n}(\mathcal{Y}_{0})\sigma\mathcal{A}_{k})$
$\displaystyle=$
$\displaystyle\mathfrak{M}(\mathcal{X}_{0}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}_{0}\sigma\mathcal{A}_{k}),$

where the limitation is taken in the sense of RTT. Therefore,
$\mathcal{X}_{0}$ is the solution of Eq. (19).

For the uniqueness of the solution $\mathcal{X}_{0}$, if both solutions
$\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ with
$\mathcal{X}_{1}\neq\mathcal{X}_{2}$ satisfy the Eq. (19), we have

$\displaystyle d_{T}(\mathcal{X}_{1},\mathcal{X}_{2})\leq\max\limits_{1\leq
i\leq
k}d_{T}(\mathcal{X}_{1}\sigma\mathcal{A}_{i},\mathcal{X}_{2}\sigma\mathcal{A}_{i})<d_{T}(\mathcal{X}_{1},\mathcal{X}_{2}),$
(22)

where Lemmas 1 and 2 are applied. Eq. (22) shows contradiction, and we should
have a unique solution of Eq. (19).

We shall prove items (III) and (IV) first. For item (III), since we assume
that $\mathcal{X}\in S_{\mbox{\tiny PD}}$ and
$\mathcal{X}\preceq\mathfrak{M}(\mathcal{X}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}\sigma\mathcal{A}_{k})$,
we have $\mathcal{X}\preceq G(\mathcal{X})\preceq\cdots$. Again, by selecting
proper $\alpha>0$, we have
$\mathcal{X},\mathcal{A}_{1},\cdots,\mathcal{A}_{n}\in S_{\alpha}$. Then, we
have $G^{n}(\mathcal{X})\preceq\alpha\mathcal{I}$ for all $n$, which implies
$G^{n}(\mathcal{X})\nearrow\mathcal{X}_{0}$ for some $\mathcal{X}_{0}\in
S_{\mbox{\tiny PD}}$ and $G(\mathcal{X}_{0})=\mathcal{X}_{0}$. From the item
(III) in the multivariate Kubo-Ando axioms given at Section (3.1), we have
$\mathcal{X}\preceq\mathcal{X}_{0}=\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
as desired.

For item (IV), since we assume that $\mathcal{X}\in S_{\mbox{\tiny PD}}$ and
$\mathcal{X}\succeq\mathfrak{M}(\mathcal{X}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}\sigma\mathcal{A}_{k})$,
we have $\mathcal{X}\succeq G(\mathcal{X})\succeq\cdots$. By selecting proper
$\alpha>0$, we have $\mathcal{X},\mathcal{A}_{1},\cdots,\mathcal{A}_{n}\in
S_{\alpha}$. Then, we have $G^{n}(\mathcal{X})\succeq\alpha^{-1}\mathcal{I}$
for all $n$, which implies $G^{n}(\mathcal{X})\searrow\mathcal{X}_{0}$ for
some $\mathcal{X}_{0}\in S_{\mbox{\tiny PD}}$ and
$G(\mathcal{X}_{0})=\mathcal{X}_{0}$. From the item (III) in the multivariate
Kubo-Ando axioms given at Section (3.1), we have
$\mathcal{X}\succeq\mathcal{X}_{0}=\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
as we wish.

For item (II), suppose we are given two sets of random tensors
$\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ and $\mathcal{B}_{i}\in
S_{\mbox{\tiny PD}}$ such that $\mathcal{A}_{i}\preceq\mathcal{B}_{i}$ for
$1\leq i\leq k$ and set $\mathcal{Y}_{0}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{B}_{1},\cdots,\mathcal{B}_{k})$.
Then we have

$\displaystyle\mathcal{Y}_{0}$ $\displaystyle=$
$\displaystyle\mathfrak{M}(\mathcal{Y}_{0}\sigma\mathcal{B}_{1},\cdots,\mathcal{Y}_{0}\sigma\mathcal{B}_{k})$
(23) $\displaystyle\succeq$
$\displaystyle\mathfrak{M}(\mathcal{Y}_{0}\sigma\mathcal{A}_{1},\cdots,\mathcal{Y}_{0}\sigma\mathcal{A}_{k})$
$\displaystyle\Longrightarrow_{1}$ $\displaystyle\mathcal{Y}_{0}$
$\displaystyle\succeq$
$\displaystyle\mathfrak{M}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$ (24)

where we use the item (IV) in this Thoerem (proved above) in
$\Longrightarrow_{1}$. Hence, we prove the item (I) at the multivariate Kubo-
Ando axioms given at Section (3.1).

Let $\mathcal{X}_{0}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$,
then from the item (II) at the multivariate Kubo-Ando axioms given at Section
(3.1), we have

$\displaystyle\mathcal{Z}^{\mathrm{H}}\mathcal{X}_{0}\mathcal{Z}$
$\displaystyle=$
$\displaystyle\mathcal{Z}^{\mathrm{H}}\mathfrak{M}(\mathcal{X}_{0}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}_{0}\sigma\mathcal{A}_{k})\mathcal{Z}$
(25) $\displaystyle=$
$\displaystyle\mathfrak{M}((\mathcal{Z}^{\mathrm{H}}\mathcal{X}_{0}\mathcal{Z})\sigma(\mathcal{Z}^{\mathrm{H}}\mathcal{A}_{1}\mathcal{Z}),\cdots,(\mathcal{Z}^{\mathrm{H}}\mathcal{X}_{0}\mathcal{Z})\sigma(\mathcal{Z}^{\mathrm{H}}\mathcal{A}_{k}\mathcal{Z}))$
$\displaystyle=$
$\displaystyle\mathfrak{M}_{\sigma}(\mathcal{Z}^{\mathrm{H}}\mathcal{A}_{1}\mathcal{Z},\cdots,\mathcal{Z}^{\mathrm{H}}\mathcal{A}_{k}\mathcal{Z}),$

where $\mathcal{Z}$ is any invertible tensor. Therefore, we prove the item
(II) at the multivariate Kubo-Ando axioms given at Section (3.1).

Let $\mathcal{A}_{i},\mathcal{A}_{i,n}\in S_{\mbox{\tiny PD}}$ and assume that
$\mathcal{A}_{i,n}\searrow\mathcal{A}_{i}$ as $n\rightarrow\infty$ for $1\leq
i\leq k$. By setting, $\mathcal{X}_{n}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1,n},\cdots,\mathcal{A}_{k,n})$,
from the fact (I) in this theorem (proved above), we have
$\mathcal{X}_{1}\succeq\mathcal{X}_{2}\succeq\cdots$ and
$\mathcal{X}_{n}\succeq\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.
Therefore, $\mathcal{X}_{n}\searrow\mathcal{X}_{0}$ for some
$\mathcal{X}_{0}\in S_{\mbox{\tiny PD}}$. Because
$\mathcal{X}_{n}\sigma\mathcal{A}_{i,n}\searrow\mathcal{X}_{0}\sigma\mathcal{A}_{i}$
for $1\leq i\leq k$, from the (III) at the multivariate Kubo-Ando axioms given
at Section (3.1) with respect to $\mathfrak{M}$, we have
$\mathcal{X}_{0}=\mathfrak{M}(\mathcal{X}_{0}\sigma\mathcal{A}_{1},\cdots,\mathcal{X}_{0}\sigma\mathcal{A}_{k})=\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.
This shows that $\mathfrak{M}_{\sigma}$ is downward continuous. On the other
hand, let $\mathcal{B}_{i},\mathcal{B}_{i,n}\in S_{\mbox{\tiny PD}}$ and
assume that $\mathcal{B}_{i,n}\nearrow\mathcal{A}_{i}$ as $n\rightarrow\infty$
for $1\leq i\leq k$. By setting,
$\mathcal{Y}_{n}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{B}_{1,n},\cdots,\mathcal{B}_{k,n})$,
from the fact (I) in this theorem (proved above), we have
$\mathcal{Y}_{1}\preceq\mathcal{Y}_{2}\preceq\cdots$ and
$\mathcal{Y}_{n}\preceq\mathfrak{M}_{\sigma}(\mathcal{B}_{1},\cdots,\mathcal{B}_{k})$.
Therefore, $\mathcal{Y}_{n}\nearrow\mathcal{Y}_{0}$ for some
$\mathcal{Y}_{0}\in S_{\mbox{\tiny PD}}$. Because
$\mathcal{Y}_{n}\sigma\mathcal{B}_{i,n}\searrow\mathcal{Y}_{0}\sigma\mathcal{Y}_{i}$
for $1\leq i\leq k$, from the (III) at the multivariate Kubo-Ando axioms given
at Section (3.1) with respect to $\mathfrak{M}$, we have
$\mathcal{Y}_{0}=\mathfrak{M}(\mathcal{Y}_{0}\sigma\mathcal{B}_{1},\cdots,\mathcal{Y}_{0}\sigma\mathcal{B}_{k})=\mathfrak{M}_{\sigma}(\mathcal{B}_{1},\cdots,\mathcal{B}_{k})$.
This shows that $\mathfrak{M}_{\sigma}$ is upward continuous. Therefore, we
prove the item (III) at the multivariate Kubo-Ando axioms given at Section
(3.1) with respect to $\mathfrak{M}_{\sigma}$.

Finally, due to that
$\mathfrak{M}(\mathcal{I}\sigma\mathcal{I},\cdots,\mathcal{I}\sigma\mathcal{I})=\mathfrak{M}_{\sigma}(\mathcal{I},\cdots,\mathcal{I})=\mathcal{I}$,
we prove the item (IV) at the multivariate Kubo-Ando axioms given at Section
(3.1) with respect to $\mathfrak{M}_{\sigma}$. $\hfill\Box$

Note that $\mathfrak{M}_{\sigma}$ in item (II) of Theorem 1 is called the
deformed mean of $\mathfrak{M}$ by $\sigma$ .

###  3.2 Examples of Multivariate Means

In this section, we will present several examples about multivariate means
that satisfy the multivariate Kubo-Ando axioms given at Section (3.1).

###### Example 1

Given a multivariate mean $\mathfrak{M}$, the _adjoin_ t of $\mathfrak{M}$,
denoted by $\mathfrak{M}^{*}$ is defined by

$\displaystyle\mathfrak{M}^{*}$ $\displaystyle=$
$\displaystyle\mathfrak{M}^{-1}(\mathcal{A}_{1}^{-1},\cdots,\mathcal{A}_{k}^{-1}),$
(26)

where $\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$. The
multivariate mean $\mathfrak{M}^{*}$ satisfies the multivariate Kubo-Ando
axioms given at Section (3.1).

###### Example 2

Given a probability vector $\bm{w}=[w_{1},\cdots,w_{k}]$ with length $k$ and
$k$ random tensors $\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq
k$, the weighted arithmetic mean with respect to $\bm{w}$, denoted as
$\mathfrak{M}_{A,\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$, can be
defined by

$\displaystyle\mathfrak{M}_{A,\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}.$ (27)

The multivariate mean
$\mathfrak{M}_{A,\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$ satisfies
the multivariate Kubo-Ando axioms given at Section (3.1).

###### Example 3

Given a probability vector $\bm{w}=[w_{1},\cdots,w_{k}]$ with length $k$ and
$k$ random tensors $\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq
k$, the weighted harmonic mean with respect to $\bm{w}$, denoted as
$\mathfrak{M}_{H,\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$, can be
defined by

$\displaystyle\mathfrak{M}_{H,\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\left(\sum\limits_{i=1}^{k}w_{i}\mathcal{A}^{-1}_{i}\right)^{-1}.$
(28)

The multivariate mean
$\mathfrak{M}_{H,\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$ satisfies
the multivariate Kubo-Ando axioms given at Section (3.1).

The bivariate power tensor mean for two PD tensors
$\mathcal{X}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ and
$\mathcal{Y}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$, represented by
$\mathcal{X}\\#_{q}\mathcal{Y}$, is defined by

$\displaystyle\mathcal{X}\\#_{q}\mathcal{Y}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathcal{X}^{1/2}\star_{N}(\mathcal{X}^{-1/2}\star_{N}\mathcal{Y}\star_{N}\mathcal{X}^{-1/2})^{q}\star_{N}\mathcal{X}^{1/2},$
(29)

where $q\in\mathbb{R}$.

###### Example 4

Given a probability vector $\bm{w}=[w_{1},\cdots,w_{k}]$ with length $k$, $k$
random tensors $\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$
and $q\in[-1,1]\backslash\\{0\\}$, the weighted power mean for $0<q\leq 1$,
represented by
$\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$, is
expressed by

$\displaystyle\mathcal{X}$ $\displaystyle=$
$\displaystyle\mathfrak{M}_{A,\bm{w}}(\mathcal{X}\\#_{q}\mathcal{A}_{1},\cdots,\mathcal{X}\\#_{q}\mathcal{A}_{k});$
(30)

and, for $-1\leq q<0$,
$\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$ is expressed
by

$\displaystyle\mathcal{X}$ $\displaystyle=$
$\displaystyle\mathfrak{M}_{H,\bm{w}}(\mathcal{X}\\#_{-q}\mathcal{A}_{1},\cdots,\mathcal{X}\\#_{-q}\mathcal{A}_{k}).$
(31)

For $0<q\leq 1$, we have
$\mathfrak{P}_{\bm{w},-q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})=\mathfrak{P}^{*}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.
The multivariate mean
$\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$ satisfies
the multivariate Kubo-Ando axioms given at Section (3.1).

###### Example 5

Given a probability vector $\bm{w}=[w_{1},\cdots,w_{k}]$ with length $k$, $k$
random tensors $\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$,
the Karcher equation is expressed by

$\displaystyle\sum\limits_{i=1}^{k}w_{i}\log(\mathcal{X}^{-1/2}\star_{N}\mathcal{A}_{i}\star_{N}\mathcal{X}^{-1/2})=\mathcal{O}.$
(32)

The unique solution to the Karcher equation above is called the Karcher mean,
represented by $\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
[18]. In [18], they show that the multivariate mean
$\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$ satisfies the
multivariate Kubo-Ando axioms given at Section (3.1). Besides, they also show
the following inequalities:

$\displaystyle\mathfrak{P}_{\bm{w},-q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\preceq\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\preceq\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(33)

where $0<q\leq 1$.

##  4 Continuity and Differentiability of Karcher Means

Theorem 6.3 in [18] shows that the Karcher mean is continuous and
differentiable in some neighborhood of the diagonal format. All tensors are
with dimensions as $\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ in this section. Given a probability vector
$\bm{w}=[w_{1},\cdots,w_{k}]$ with length $k$ and $k$ identity tensors
$\mathcal{I}$, the following map

$\displaystyle(\underbrace{\mathcal{I},\cdots,\mathcal{I}}_{k\leavevmode\nobreak\
\mbox{terms}})\rightarrow\mathfrak{G}_{\bm{w}}(\underbrace{\mathcal{I},\cdots,\mathcal{I}}_{k\leavevmode\nobreak\
\mbox{terms}}),$ (34)

is continuous and differentiable in some neighborhood of
$(\underbrace{\mathcal{I},\cdots,\mathcal{I}}_{k\leavevmode\nobreak\
\mbox{terms}})$. In this section, we wish to extend this diagonal format to
the following general format, i.e., the following map

$\displaystyle(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\rightarrow\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(35)

is continuous and differentiable in some neighborhood of any
$(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\in S^{k}_{\mbox{\tiny PD}}$ in the
sense of RTT. The main technique used in this section is based on
_prederiivative_ of the function introduced in [24].

Let $\mathrm{X},\mathrm{Y}$ be two Hilbert spaces, and $\mathfrak{A}$ be a
collection of all random linear tensors (operators) from the space
$\mathrm{X}$ to the space $\mathrm{Y}$. The noncompactness measure of
$\mathfrak{A}$, denoted as $\beta(\mathfrak{A})$, is defind by

$\displaystyle\beta(\mathfrak{A})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\inf\\{r|\exists
k\in\mathbb{N},\exists\mathcal{A}_{1},\cdots,\mathcal{A}_{k}\in\mathfrak{A}\leavevmode\nobreak\
\mbox{such that}\leavevmode\nobreak\
\mathfrak{A}\subset\bigcup\limits_{i=1}^{k}B(A_{i},r)\\}.$ (36)

If $\mathfrak{A}$ is a compact space, $\beta(\mathfrak{A})=0$. The other
measure is about the surjectivity measure of $\mathcal{A}$, denoted as
$\gamma(\mathfrak{A})$, which is defined by

$\displaystyle\gamma(\mathcal{A})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sup\\{r\in\mathbb{R}\leavevmode\nobreak\ \mbox{such
that}\leavevmode\nobreak\
r\mathbb{B}_{\mathrm{Y}}\subset\mathcal{A}(\mathbb{B}_{\mathrm{X}})\\},$ (37)

where $\mathcal{A}\in\mathfrak{A}$, and
$\mathbb{B}_{\mathrm{X}},\mathbb{B}_{\mathrm{Y}}$ are any unit norm balls
inside spaces of $\mathrm{X},\mathrm{Y}$, respectively. Then,
$\gamma(\mathfrak{A})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\inf\limits_{\mathcal{A}\in\mathfrak{A}}\gamma(\mathcal{A})$.

Given two PD tensors $\mathcal{X}^{\prime}$ and $\mathcal{X}^{\prime\prime}$,
we say that these two tensors are _square root representable_ if we have

$\displaystyle\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}$
$\displaystyle=$ $\displaystyle(\mathcal{X}^{\prime
1/2}-\mathcal{X}^{\prime\prime 1/2})(\mathcal{X}^{\prime
1/2}+\mathcal{X}^{\prime\prime
1/2}+\mathcal{C}_{\mathcal{X}^{\prime},\mathcal{X}^{\prime\prime}}),$ (38)

where
$\mathcal{C}_{\mathcal{X}^{\prime},\mathcal{X}^{\prime\prime}}\in\mathfrak{A}$.
Note that the tensor
$\mathcal{C}_{\mathcal{X}^{\prime},\mathcal{X}^{\prime\prime}}\in\mathfrak{A}=\mathcal{O}$
if
$\mathcal{X}^{\prime}\star_{N}\mathcal{X}^{\prime\prime}=\mathcal{X}^{\prime\prime}\star_{N}\mathcal{X}^{\prime}$,
i.e., commutative.

###### Lemma 3

Let $D\subset S_{\mbox{\tiny PD}}$ be an open set that includes
$\mathfrak{G}_{\bm{w}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})\in
D$ and all pair of distinct tensors within $D$ are square root representable
defined by Eq. (38), where
$\mathfrak{G}_{\bm{w}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})$ is
the Karcher mean with respect to the weight $\bm{w}$ and
$(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})\in S^{k}_{\mbox{\tiny
PD}}$. We define the following tensor-valued function:

$\displaystyle
F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sum\limits_{i=1}^{k}w_{i}\log(\mathcal{X}^{1/2}\star_{N}\mathcal{A}^{-1}_{i}\star_{N}\mathcal{X}^{1/2}).$
(39)

Then, we have the following property with respect to the function
$F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X})$. For all
$\epsilon>0$, there exists a neighborhood $W_{\epsilon}\subset
S^{k}_{\mbox{\tiny PD}}\times D$ of
$(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k},\mathfrak{G}_{\bm{w}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k}))$
such that, for any
$(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X}^{\prime}),(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X}^{\prime\prime})\in
W_{\epsilon}$, we have:

$\displaystyle
F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X}^{\prime})-F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X}^{\prime\prime})$
$\displaystyle=$
$\displaystyle\acute{\mathcal{A}}\star_{N}(\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime})+\epsilon\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}\mathcal{B}$
(40)

where $\acute{\mathcal{A}},\mathcal{B}\in\mathfrak{A}$, and $\mathcal{B}$ is a
tensor with norm $\left\|\mathcal{B}\right\|_{\rho}\leq 1$.

Proof: In the region $D$ around
$\mathfrak{G}_{\bm{w}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})$,
we select a neighborhood $D_{\delta}$ of
$\mathcal{X}^{\prime},\mathcal{X}^{\prime\prime}$ such that

$\displaystyle\mathcal{X}^{\prime 1/2}=\mathcal{X}^{\prime\prime 1/2}+\Delta,$
(41)

where the tensor $\Delta$ satisfies with $\left\|\Delta\right\|_{\rho}<\delta$
with $\delta$ having

$\displaystyle\delta^{2}\leq
C\epsilon\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho},$
(42)

where $C>0$ is a constant indepdent of selection $\mathcal{X}^{\prime}$ and
$\mathcal{X}^{\prime\prime}$ in $D$.

and $\delta$. From the definition provided by Eq. (39), we have

$\displaystyle
F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X}^{\prime})-F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X}^{\prime\prime})$
(43) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{k}w_{i}\log\frac{(\mathcal{X}^{\prime\prime
1/2}+\Delta)\mathcal{A}^{-1}_{i}(\mathcal{X}^{\prime\prime
1/2}+\Delta)}{\mathcal{X}^{\prime\prime-1/2}\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime-1/2}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{k}w_{i}\log\left(\mathcal{I}+\frac{\Delta\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime
1/2}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}+\frac{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}^{-1}_{i}\Delta}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}+\frac{\Delta\mathcal{A}^{-1}_{i}\Delta}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime 1/2}}\right)$
$\displaystyle=_{1}$
$\displaystyle\sum\limits_{i=1}^{k}w_{i}\left(\widetilde{\frac{\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime
1/2}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}}+\frac{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}^{-1}_{i}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}\right)\Delta+\mathrm{O}(\Delta^{2})$ $\displaystyle=_{2}$
$\displaystyle\sum\limits_{i=1}^{k}w_{i}\left(\widetilde{\frac{\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime
1/2}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}}+\frac{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}^{-1}_{i}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}\right)\frac{(\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime})}{(\mathcal{X}^{\prime
1/2}+\mathcal{X}^{\prime\prime
1/2}+\mathcal{C}_{\mathcal{X}^{\prime},\mathcal{X}^{\prime\prime}})}$
$\displaystyle+\mathrm{O}(\Delta^{2})$ $\displaystyle=_{3}$
$\displaystyle\acute{\mathcal{A}}(\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime})+\epsilon\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}\mathcal{B}$

where $=_{1}$ comes from the Taylor expansion for
$\log(1+x)=\sum\limits_{i=1}^{\infty}(-1)^{i-1}\frac{x^{i}}{i}$ and
$\frac{\Delta\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime
1/2}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}=\widetilde{\frac{\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime
1/2}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime 1/2}}}\Delta$, $=_{2}$ uses
the fact that tensors $\mathcal{X}^{\prime}$ and $\mathcal{X}^{\prime\prime}$
are _square root representable_ , and $=_{3}$ comes from Eq. (42) and summands
rearrangments by setting

$\displaystyle\acute{\mathcal{A}}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\left[\sum\limits_{i=1}^{k}w_{i}\left(\widetilde{\frac{\mathcal{A}^{-1}_{i}\mathcal{X}^{\prime\prime
1/2}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}}+\frac{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}^{-1}_{i}}{\mathcal{X}^{\prime\prime
1/2}\mathcal{A}_{i}^{-1}\mathcal{X}^{\prime\prime
1/2}}\right)\right]\left(\mathcal{X}^{\prime 1/2}+\mathcal{X}^{\prime\prime
1/2}+\mathcal{C}_{\mathcal{X}^{\prime},\mathcal{X}^{\prime\prime}}\right)^{-1}.$
(44)

Then, this Lemma is proved by selecting $W_{\epsilon}=\tilde{S}_{\mbox{\tiny
PD,1}}\times\cdots\times\tilde{S}_{\mbox{\tiny PD,k}}\times D_{\delta}$, where
$\tilde{S}_{\mbox{\tiny PD,i}}$ is the open set in $S_{\mbox{\tiny PD}}$ that
contains $\hat{\mathcal{A}}_{i}$ for $1\leq i\leq k$. $\hfill\Box$

Following Lemma is about the implicit function of
$F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X})$ if Lemma 3 is
satisfied. We define $\mathfrak{C}(\mathrm{X},\mathrm{Y})$ as a collection of
all continuous functions mapping from the space $\mathrm{X}$ to the space
$\mathrm{Y}$.

###### Lemma 4

Let $D\subset S_{\mbox{\tiny PD}}$ be an open set that includes
$\mathfrak{G}_{\bm{w}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})\in
D$ and all pair of distinct tensors within $D$ are square root representable
defined by Eq. (38), where
$\mathfrak{G}_{\bm{w}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})$ is
the Karcher mean with respect to the weight $\bm{w}$ and
$(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})\in S^{k}_{\mbox{\tiny
PD}}$. We define the following tensor-valued function:

$\displaystyle
F_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k},\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sum\limits_{i=1}^{k}w_{i}\log(\mathcal{X}^{1/2}\star_{N}\mathcal{A}^{-1}_{i}\star_{N}\mathcal{X}^{1/2}).$
(45)

For notation simplicity, we set
$\underline{\hat{\mathcal{A}}}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})$. Then, for
all $\beta(\mathfrak{A})<\tau_{1}<\tau_{2}<\gamma(\mathfrak{A})$, there exists
a neighborhood $U$ of
$(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}}),F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})))$
and a function $\phi:U\rightarrow D$ such that

$\displaystyle
F_{\bm{w}}(\underline{\mathcal{A}},\phi(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y}))$
$\displaystyle=$ $\displaystyle\mathcal{Y},$ (46)

and

$\displaystyle\left\|\mathcal{X}-\phi(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y})\right\|_{\rho}$
$\displaystyle\leq$
$\displaystyle\frac{\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X})-\mathcal{Y}\right\|_{\rho}}{\tau_{2}-\tau_{1}},$
(47)

are satisfied by all $(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y})\in U$.

Proof: Pick a positive number $r$ betwee $\beta(\mathfrak{A})$ and $\tau_{1}$
and the definition of $\beta(\mathfrak{A})$ provied by Eq. (36), we have

$\displaystyle\mathfrak{A}\subset\bigcup\limits_{i=1}^{k}B(\mathcal{A}_{i},r).$
(48)

Let $\Upsilon_{\underline{\mathcal{A}}}$ be the convex hull of
$\\{\mathcal{A}_{1},\cdots,\mathcal{A}_{k}\\}$. Then
$\Upsilon_{\underline{\mathcal{A}}}\in\mathfrak{A}$ is compact and convex. By
setting $\epsilon_{1}=\tau_{1}-r$, from Lemma 3, we have a neighborhood
$W_{\epsilon_{1}}\subset S^{k}_{\mbox{\tiny PD}}\times D$ such that

$\displaystyle
F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime})-F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime})$
$\displaystyle=$
$\displaystyle\acute{\mathcal{A}}(\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime})+\epsilon_{1}\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}\mathcal{B},$
(49)

where
$(\underline{\mathcal{A}},\mathcal{X}^{\prime}),(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime})\in
W_{\epsilon_{1}}$. Note that
$\mathfrak{A}\subset\Upsilon_{\underline{\mathcal{A}}}+r\mathbb{B}_{L(S_{\mbox{\tiny
PD}},S_{\mbox{\tiny PD}})}$, where $\mathbb{B}_{L(S_{\mbox{\tiny
PD}},S_{\mbox{\tiny PD}})}$ is the set of all tensors (opertors) from
$S_{\mbox{\tiny PD}}$ to $S_{\mbox{\tiny PD}}$ with the norm measured by
$\left\|\cdot\right\|_{\rho}$ less or equal then one.

Pick any $\upsilon\in\Upsilon_{\underline{\mathcal{A}}}$, since the mapping
$\upsilon\rightarrow\gamma(\upsilon)$ is continuous on
$\Upsilon_{\underline{\mathcal{A}}}$ and the set
$\Upsilon_{\underline{\mathcal{A}}}$ is compact, we can find a constant $c$
such that $\tau_{2}<c<\gamma(\upsilon)$ for all
$\upsilon\in\Upsilon_{\underline{\mathcal{A}}}$. We can have a surjective map
$g:\mathbb{C}(\Upsilon_{\underline{\mathcal{A}}},S_{\mbox{\tiny
PD}})\rightarrow\mathbb{C}(\Upsilon_{\underline{\mathcal{A}}},S_{\mbox{\tiny
PD}})$ such that

$\displaystyle
c\mathbb{B}_{\mathbb{C}(\Upsilon_{\underline{\mathcal{A}}},S_{\mbox{\tiny
PD}})}\subset
g(\mathbb{B}_{\mathbb{C}(\Upsilon_{\underline{\mathcal{A}}},S_{\mbox{\tiny
PD}})}).$ (50)

where $0<c<\gamma(g)$.

Set $\epsilon_{2}=\tau_{2}-\tau_{1}$ and select $\delta>0$ with a neighborhood
$U$ of
$(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}}),F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})))$
such that

$\displaystyle(\underline{\mathcal{A}},\mathcal{T})\in
W_{\epsilon_{1}},\leavevmode\nobreak\
\left\|\mathcal{X}-\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})\right\|_{\rho}\leq\frac{\delta}{3},\leavevmode\nobreak\
\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X})-\mathcal{Y}\right\|_{\rho}\leq\frac{\epsilon_{2}\delta}{3},$
(51)

where $(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y})\in U$ and
$\mathcal{T}\in\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})+\delta\mathbb{B}_{S_{\mbox{\tiny
PD}}}$.

We will apply Ekeland’s variational principle to obtain the function $\phi$
that satisfies Eqs. (46) and (47) [25]. By applying Ekeland’s variational
principle to the complete metric space
$\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})+\delta\mathbb{B}_{S_{\mbox{\tiny
PD}}}$ and the mapping
$\mathcal{T}\rightarrow\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{T})-\mathcal{Y}\right\|_{\rho}$,
there exists the function
$\phi\in\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})+\delta\mathbb{B}_{S_{\mbox{\tiny
PD}}}$ satisfying the following relations:

$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\mathcal{Y}\right\|_{\rho}+\epsilon_{2}\left\|\mathcal{X}-\phi\right\|_{\rho}$
$\displaystyle\leq$
$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X})-\mathcal{Y}\right\|_{\rho},$
(52)

and

$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\mathcal{Y}\right\|_{\rho}$
$\displaystyle\leq$
$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{T})-\mathcal{Y}\right\|_{\rho}+\epsilon_{2}\left\|\mathcal{T}-\phi\right\|_{\rho},$
(53)

where
$\mathcal{T}\in\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})+\delta\mathbb{B}_{S_{\mbox{\tiny
PD}}}$. Then, the inequality provided by Eq. (52) implies Eq. (47). Now, we
want to prove Eq. (46).

From Eq. (52), we have

$\displaystyle\left\|\mathcal{X}-\phi\right\|_{\rho}\leq\frac{\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X})-\mathcal{Y}\right\|_{\rho}}{\epsilon_{2}}\leq\frac{\delta}{3},$
(54)

combined with the condition
$\left\|\mathcal{X}-\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})\right\|_{\rho}\leq\frac{\delta}{3}$,
we obtain

$\displaystyle\left\|\phi-\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})\right\|_{\rho}\leq\frac{2\delta}{3}.$
(55)

From Eq. (51), there exists an element $\mathcal{X}_{\upsilon}$ for all
$\epsilon\in\Upsilon_{\underline{\mathcal{A}}}$ such that the map
$\upsilon\rightarrow\mathcal{X}_{\upsilon}$ is continuous and the following
relations:

$\displaystyle\mathcal{Y}-F_{\bm{w}}(\underline{\mathcal{A}},\phi)$
$\displaystyle=$ $\displaystyle\upsilon\mathcal{X}_{\upsilon},$ (56)

and

$\displaystyle\left\|\mathcal{X}_{\upsilon}\right\|_{\rho}$ $\displaystyle<$
$\displaystyle\frac{\delta}{3},$ (57)

where $\upsilon\in\Upsilon_{\underline{\mathcal{A}}}$. If we set
$\mathcal{T}=\phi+\mathcal{X}_{\upsilon}$, from Eq. (55) and Eq. (57), we have
$\mathcal{T}\in\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})+\delta\mathbb{B}_{S_{\mbox{\tiny
PD}}}$, then, from Eq. (53), we have

$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\mathcal{Y}\right\|_{\rho}$
$\displaystyle\leq$
$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi+\mathcal{X}_{\upsilon})-F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\upsilon\mathcal{X}_{\upsilon}\right\|_{\rho}+\epsilon_{2}\left\|\mathcal{X}_{\upsilon}\right\|_{\rho},$
(58)

where $\upsilon\in\Upsilon_{\underline{\mathcal{A}}}$.

We define the set-valued map
$\Phi:\Upsilon_{\underline{\mathcal{A}}}\rightarrow
2^{\Upsilon_{\underline{\mathcal{A}}}}$ by

$\displaystyle\Phi(\upsilon)\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{\varrho\in\Upsilon_{\underline{\mathcal{A}}}|\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi+\mathcal{X}_{\upsilon})-F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\varrho\mathcal{X}_{\upsilon}\right\|_{\rho}\leq\tau_{1}\left\|\mathcal{X}_{\upsilon}\right\|_{\rho}\\}.$
(59)

From Lemma 3, we know that the set $\Phi(\upsilon)$ is nonempty for all
$\upsilon\in\Upsilon_{\underline{\mathcal{A}}}$. Besides, the set
$\Phi(\upsilon)$ is a convex and compact set. Before applying Kakutani’s fixed
point theorem to the set $\Phi(\upsilon)$, we have to show that the set
$\Phi(\upsilon)$ is semicontinuous. Let
$\upsilon_{0}\in\Upsilon_{\underline{\mathcal{A}}}$ be an arbitrary element,
$\upsilon_{n},\varrho_{n}\in\Phi(\upsilon_{n})$ are two sequences with
$\upsilon_{0}=\lim\limits_{n\rightarrow\infty}\upsilon_{n}$ and
$\varrho_{0}=\lim\limits_{n\rightarrow\infty}\varrho_{n}$. Since we have

$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi+\mathcal{X}_{\upsilon_{n}})-F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\varrho_{n}\mathcal{X}_{\upsilon_{n}}\right\|_{\rho}\leq\tau_{1}\left\|\mathcal{X}_{\upsilon_{n}}\right\|_{\rho},$
(60)

by taking $n\rightarrow\infty$ and using the continuity of
$\upsilon\rightarrow\mathcal{X}_{\upsilon}$, we have
$\varrho_{0}\in\Phi(\upsilon_{0})$. This shows that the set $\Phi(\upsilon)$
is semicontinuous.

From Kakutani’s fixed point theorem, there exists a fixed element
$\tilde{\upsilon}\in\Phi(\upsilon)$, i.e.,

$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi+\mathcal{X}_{\tilde{\upsilon}})-F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\tilde{\upsilon}\mathcal{X}_{\tilde{\upsilon}}\right\|_{\rho}\leq\tau_{1}\left\|\mathcal{X}_{\tilde{\upsilon}}\right\|_{\rho}.$
(61)

By setting $\upsilon=\tilde{\upsilon}$ into Eq. (58) and applying the
inequality provided by Eq. (57), we get

$\displaystyle\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\mathcal{Y}\right\|_{\rho}\leq(\tau_{1}+\epsilon_{2})\left\|\mathcal{X}_{\tilde{\upsilon}}\right\|_{\rho}=\tau_{2}\left\|\mathcal{X}_{\tilde{\upsilon}}\right\|_{\rho}\leq\frac{\tau_{2}\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\mathcal{Y}\right\|_{\rho}}{c}.$
(62)

Because $\tau_{2}<c$, the inequality given by Eq. (62) yields
$\left\|F_{\bm{w}}(\underline{\mathcal{A}},\phi)-\mathcal{Y}\right\|_{\rho}=0$.
This shows the existence of $\phi$ that satifies Eq. (46). $\hfill\Box$

Before presenting our main theorem in this section about the continuity and
the differentiability of Karcher Means, we need to present one more lemma
about the surjective measure $\gamma$.

###### Lemma 5

Let $\mathrm{X},\mathrm{Y}$ be two Hilbert spaces of tensors and
$\mathcal{A}:\mathrm{X}\rightarrow\mathrm{Y}$ be an injective bouded tensor,
we have

$\displaystyle\left\|\mathcal{A}\mathcal{X}\right\|_{\rho}\geq\gamma(\mathcal{A})\left\|\mathcal{X}\right\|_{\rho},$
(63)

where $\forall\mathcal{X}\in\mathrm{X}$.

Proof: If $\gamma(\mathcal{A})=0$, this makes Eq. (63) valid. We will prove
this Lemma by contradiction. Suppose Eq. (63) is not valid for
$\gamma(\mathcal{A})>0$, then there exists a positive number
$c<\gamma(\mathcal{A})$ and an $\mathcal{X}\in\mathrm{X}$ such that

$\displaystyle\left\|\mathcal{A}\mathcal{X}\right\|_{\rho}<c\left\|\mathcal{X}\right\|_{\rho}.$
(64)

Then, we have

$\displaystyle\frac{c}{\left\|\mathcal{A}\right\|_{\rho}}\mathcal{A}\mathcal{X}\in
c\mathbb{B}_{\mathrm{Y}}\subset\mathcal{A}\mathbb{B}_{\mathrm{X}},$ (65)

which is equivalent to have $\tilde{\mathcal{X}}$ such that

$\displaystyle\mathcal{A}\left(\frac{c}{\left\|\mathcal{A}\mathcal{X}\right\|_{\rho}}\mathcal{X}\right)=\mathcal{A}(\tilde{\mathcal{X}}).$
(66)

From the injectivity of $\mathcal{A}$, we have

$\displaystyle\frac{c}{\left\|\mathcal{A}\mathcal{X}\right\|_{\rho}}\mathcal{X}$
$\displaystyle=$ $\displaystyle\tilde{\mathcal{X}}.$ (67)

By taking the $\left\|\cdot\right\|_{\rho}$ operation at both sides of Eq.
(68), we have

$\displaystyle
1<\frac{c\left\|\mathcal{X}\right\|_{\rho}}{\left\|\mathcal{A}\mathcal{X}\right\|_{\rho}}$
$\displaystyle=$ $\displaystyle\left\|\tilde{\mathcal{X}}\right\|_{\rho}\leq
1.$ (68)

However, this contradics to our assumption. $\hfill\Box$

We are ready to present our main theorem about the continuity and the
differentiability of Karcher Means in this section. For notational simplicity,
we set $\underline{\hat{\mathcal{A}}}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}(\hat{\mathcal{A}}_{1},\cdots,\hat{\mathcal{A}}_{k})$.

###### Theorem 2

Let $D\subset S_{\mbox{\tiny PD}}$ be an open set that includes
$\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})\in D$ and all pair of
distinct tensors within $D$ are square root representable defined by Eq. (38),
where $\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})$ is the Karcher
mean with respect to the weight $\bm{w}$ and $\underline{\hat{\mathcal{A}}}\in
S^{k}_{\mbox{\tiny PD}}$. We define the following tensor-valued function:

$\displaystyle
F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sum\limits_{i=1}^{k}w_{i}\log(\mathcal{X}^{1/2}\star_{N}\mathcal{A}^{-1}_{i}\star_{N}\mathcal{X}^{1/2}).$
(69)

We assume that all tensors from $\mathfrak{A}$, i.e., a collection of all
random linear tensors from the space $S_{\mbox{\tiny PD}}$ to the space
$S_{\mbox{\tiny PD}}$, are injective. Then, there exist a neighborhood $V$ of
$(\underline{\hat{\mathcal{A}}},F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})))$
and a unique determined function $\psi:V\rightarrow D$ such that

$\displaystyle
F_{\bm{w}}(\underline{\mathcal{A}},\psi(\underline{\mathcal{A}},\mathcal{Y}))=\mathcal{Y},$
(70)

where $(\underline{\mathcal{A}},\mathcal{Y})\in V$. The mapping below

$\displaystyle\underline{\mathcal{A}}$ $\displaystyle\rightarrow$
$\displaystyle\psi(\underline{\mathcal{A}},F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}}))),$
(71)

is continuous at $\underline{\mathcal{A}}=\underline{\hat{\mathcal{A}}}$.
Finally, the mapping $\psi(\underline{\mathcal{A}},\mathcal{Y})$ is
prederivative, i.e., for all $\epsilon>0$, there exists a neighborhood
$V_{\epsilon}\subset V$ such that

$\displaystyle\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})=\acute{\mathcal{A}}^{-1}\star_{N}(\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime})+\epsilon\left\|\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}\right\|_{\rho}\mathcal{B},$
(72)

where $\acute{\mathcal{A}}\in\mathfrak{A}$ and
$(\underline{\mathcal{A}},\mathcal{Y}^{\prime}),(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})\in
V_{\epsilon}$.

Proof: Let $\beta(\mathfrak{A})<r<\tau_{1}<\tau_{2}<\gamma(\mathfrak{A})$,
$\epsilon_{1}=\tau_{1}-r<\gamma(\mathfrak{A})$, and $W=W_{\epsilon_{1}}$ be
the corresponding nieghborhood in Lemma 4. Our first goal is to show that if
$F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime})=F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime})$,
this implies $\mathcal{X}^{\prime}=\mathcal{X}^{\prime\prime}$. If
$F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime})=F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime})$,
then, from Lemma 3, there exists $\mathcal{A}\in\mathfrak{A}$ and
$\mathcal{Y}$ with $\left\|\mathcal{Y}\right\|_{\rho}\leq 1$ such that

$\displaystyle\mathcal{A}(\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime})+\epsilon_{1}\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}\mathcal{Y}=\mathcal{O}.$
(73)

From Lemma 5, we have

$\displaystyle\epsilon_{1}\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}$
$\displaystyle\geq$
$\displaystyle\epsilon_{1}\left\|\mathcal{Y}\right\|_{\rho}\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}$
(74) $\displaystyle=$
$\displaystyle\left\|\mathcal{A}(\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime})\right\|_{\rho}$
$\displaystyle\geq$
$\displaystyle\gamma(\mathcal{A})\left\|\mathcal{X}^{\prime}-\mathcal{X}^{\prime\prime}\right\|_{\rho}.$

If $\mathcal{X}^{\prime}\neq\mathcal{X}^{\prime\prime}$, we will obtain a
contradiction as $\epsilon_{1}\geq\gamma(\mathcal{A})$. Therefore, if
$F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime})=F_{\bm{w}}(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime})$,
we have $\mathcal{X}^{\prime}=\mathcal{X}^{\prime\prime}$.

From Lemma 4, there exists a neighborhood $U$ and $\phi$ such that
$(\underline{\mathcal{A}},\phi(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y}))\in
W_{\epsilon_{1}}$ given $(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y})\in
U$. Define $V$ as the projection of $U$ on the space $S^{k}_{\mbox{\tiny
PD}}\times S_{\mbox{\tiny PD}}$ and define $\psi:V\rightarrow D$ by

$\displaystyle\psi(\underline{\mathcal{A}},\mathcal{Y})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\phi(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y}),$ (75)

where $(\underline{\mathcal{A}},\mathcal{X},\mathcal{Y})\in U$. If
$(\underline{\mathcal{A}},\mathcal{X}^{\prime},\mathcal{Y})$ and
$(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime},\mathcal{Y})$ are in $U$,
then
$(\underline{\mathcal{A}},\phi(\underline{\mathcal{A}},\mathcal{X}^{\prime},\mathcal{Y}))$
and
$(\underline{\mathcal{A}},\phi(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime},\mathcal{Y}))$
are in $W_{\epsilon_{1}}$. From Eq. (46), we have

$\displaystyle
F_{\bm{w}}(\underline{\mathcal{A}},\phi(\underline{\mathcal{A}},\mathcal{X}^{\prime},\mathcal{Y}))=\mathcal{Y}=F_{\bm{w}}(\underline{\mathcal{A}},\phi(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime},\mathcal{Y})).$
(76)

We have Eq. (70) from
$\phi(\underline{\mathcal{A}},\mathcal{X}^{\prime},\mathcal{Y})=\phi(\underline{\mathcal{A}},\mathcal{X}^{\prime\prime},\mathcal{Y})$.

By setting $\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})$ and
$\mathcal{Y}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}}))$
into the inequality provided by Eq. (47), we have

$\displaystyle\left\|\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})-\psi(\underline{\mathcal{A}},F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}})))\right\|_{\rho}\leq\frac{\left\|F_{\bm{w}}(\underline{\mathcal{A}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}}))-F_{\bm{w}}(\underline{\hat{\mathcal{A}}},\mathfrak{G}_{\bm{w}}(\underline{\hat{\mathcal{A}}}))\right\|_{\rho}}{\tau_{2}-\tau_{1}}.$
(77)

As R.H.S. of Eq. (77) tends to zero as $\underline{\mathcal{A}}$ tends to
$\underline{\hat{\mathcal{A}}}$, the continuity of Eq. (71) is proved.

The last piece of this proof is to show Eq. (72). By setting
$\mathcal{Y}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathcal{Y}^{\prime\prime}$ and
$\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})$ into the
inequality provided by Eq. (47), we obtain

$\displaystyle\left\|\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})\right\|_{\rho}\leq\frac{\left\|\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}\right\|_{\rho}}{\tau_{2}-\tau_{1}},$
(78)

where
$(\underline{\mathcal{A}},\mathcal{Y}^{\prime}),(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})\in
V$. Choose $V_{\epsilon}\subset V$ so that
$(\underline{\mathcal{A}},\mathcal{Y})\in V_{\epsilon}$ implies
$(\underline{\mathcal{A}},\psi(\underline{\mathcal{A}},\mathcal{Y}))\in
W_{\epsilon\tau_{2}(\tau_{2}-\tau_{1})}$, where $W_{\epsilon}$ is the open set
required to have Eq. (40) in Lemma 3. By selecting two elements
$(\underline{\mathcal{A}},\mathcal{Y}^{\prime}),(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})\in
V_{\epsilon}$, and setting
$\mathcal{X}^{\prime}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime}),\mathcal{X}^{\prime\prime}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})$ in Eq.
(40) of Lemma 3, we have

$\displaystyle\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}$
$\displaystyle=$
$\displaystyle\acute{\mathcal{A}}(\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime}))+\epsilon\tau_{2}(\tau_{2}-\tau_{1})\left\|\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})\right\|_{\rho}\mathcal{B}_{S_{\mbox{\tiny
PD}}}$ (79) $\displaystyle\subset_{1}$
$\displaystyle\acute{\mathcal{A}}(\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime}))+\epsilon\tau_{2}\left\|\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}\right\|_{\rho}\mathcal{B}_{S_{\mbox{\tiny
PD}}},$

where $\mathcal{B}_{S_{\mbox{\tiny PD}}}$ is a bounded norm (by 1) tensor in
the space $S_{\mbox{\tiny PD}}$ and $\subset_{1}$ is obtained by applying the
relation provided by Eq. (78). This means that we can find an tenor
$\acute{\acute{\mathcal{A}}}\in\mathfrak{A}$, such that

$\displaystyle\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}$
$\displaystyle=$
$\displaystyle\acute{\acute{\mathcal{A}}}(\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime}))+\epsilon\tau_{2}\left\|\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}\right\|_{\rho}\mathcal{B}_{S_{\mbox{\tiny
PD}}}.$ (80)

By taking the inverse of $\acute{\acute{\mathcal{A}}}$ (since all tensors in
$\mathfrak{A}$ are injective) in Eq. (80) and using the fact that
$\acute{\acute{\mathcal{A}}}^{-1}\mathbb{B}_{S_{\mbox{\tiny
PD}}}\subset\frac{1}{\tau_{2}}\mathbb{B}_{S_{\mbox{\tiny PD}}}$, we have

$\displaystyle\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime})-\psi(\underline{\mathcal{A}},\mathcal{Y}^{\prime\prime})=\acute{\acute{\mathcal{A}}}^{-1}\star_{N}(\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime})+\epsilon\left\|\mathcal{Y}^{\prime}-\mathcal{Y}^{\prime\prime}\right\|_{\rho}\mathcal{B}.$
(81)

$\hfill\Box$

##  5 Tail bounds for Multivariate Random Tensor with Power Mean and Karcher
Mean

In this section, we will establish tail bounds for multivariate random tensors
with power mean and Karcher mean. Following lemmas are required to prove the
main results of this section.

###### Lemma 6

Let $\mathfrak{M}:S^{k}_{\mbox{\tiny PD}}\rightarrow S_{\mbox{\tiny PD}}$ be a
map of _$k$ -variable tensor mean_ and $\sigma$ is a p.m.i. operator mean. If
we have

$\displaystyle\mathfrak{M}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\succeq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(82)

where $p\geq 1$, then,

$\displaystyle\mathfrak{M}_{\sigma}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\succeq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$
(83)

On ther other hand, if we have

$\displaystyle\mathfrak{M}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(84)

where $0<p\leq 1$, then,

$\displaystyle\mathfrak{M}_{\sigma}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$
(85)

Proof: We will prove Eq. (83) first. We begin with the assumption that $1\leq
p\leq 2$, define $\mathcal{X}$ as
$\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$, and
set $\acute{\lambda}$ by $\acute{\lambda}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\lambda_{\min}(\mathcal{X})$. From the congruence invaraince
provided by Eq. (9), we have

$\displaystyle\mathcal{I}=\mathfrak{M}(g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{1}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{k}\mathcal{X}^{-1/2})).$
(86)

From (III) in Theorem 1, it is enough to prove

$\displaystyle\acute{\lambda}^{p-1}\mathcal{X}$ $\displaystyle\preceq$
$\displaystyle\mathfrak{M}((\acute{\lambda}^{p-1}\mathcal{X})\sigma\mathcal{A}_{1}^{p},\cdots,(\acute{\lambda}^{p-1}\mathcal{X})\sigma\mathcal{A}_{k}^{p}),$
(87)

and, from congruence invariance, this is equivalent to prove

$\displaystyle\mathcal{I}$ $\displaystyle\preceq$
$\displaystyle\mathfrak{M}(g_{\sigma}(\acute{\lambda}^{1-p}\mathcal{X}^{-1/2}\mathcal{A}_{1}^{p}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\acute{\lambda}^{1-p}\mathcal{X}^{-1/2}\mathcal{A}_{k}^{p}\mathcal{X}^{-1/2})).$
(88)

From Hansen-Pedersen’s inequality, for $1\leq i\leq k$, we have

$\displaystyle
g_{\sigma}(\acute{\lambda}^{1-p}\mathcal{X}^{-1/2}\mathcal{A}_{i}^{p}\mathcal{X}^{-1/2})\succeq
g_{\sigma}((\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2})^{p})\succeq_{1}g^{p}_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2}),$
(89)

where $\succeq_{1}$ comes from p.m.i. of $\sigma$. From the assumption of Eq.
(82), we have

$\displaystyle\mathfrak{M}(g_{\sigma}(\acute{\lambda}^{1-p}\mathcal{X}^{-1/2}\mathcal{A}_{1}^{p}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\acute{\lambda}^{1-p}\mathcal{X}^{-1/2}\mathcal{A}_{k}^{p}\mathcal{X}^{-1/2}))$
(90)
$\displaystyle\succeq\mathfrak{M}(g^{p}_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{1}\mathcal{X}^{-1/2}),\cdots,g^{p}_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{k}\mathcal{X}^{-1/2}))$
$\displaystyle\succeq\lambda_{\min}^{p-1}\left(\mathfrak{M}(g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{1}^{p}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{k}^{p}\mathcal{X}^{-1/2}))\right)$
$\displaystyle\times\mathfrak{M}(g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{1}^{p}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{k}^{p}\mathcal{X}^{-1/2}))\succeq\lambda_{\min}^{p-1}(\mathcal{I})\mathcal{I}=\mathcal{I}.$

Therefore, we prove Eq. (83) for $1\leq p\leq 2$. For $p\geq 2$, we willuse
induction. Suppose we have Eq. (83) for $1\leq p\leq 2^{m}$, for $2^{m}\leq
p\leq 2^{m+1}$, we can express $p=2p^{\prime}$ with $1\leq p^{\prime}\leq
2^{m}$. Then,

$\displaystyle\mathfrak{M}_{\sigma}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\succeq$
$\displaystyle\lambda_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1}^{p^{\prime}},\cdots,\mathcal{A}_{k}^{p^{\prime}}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1}^{p^{\prime}},\cdots,\mathcal{A}_{k}^{p^{\prime}})$
(91) $\displaystyle\succeq$
$\displaystyle\lambda_{\min}(\lambda^{p^{\prime}-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))$
$\displaystyle\times\lambda^{p^{\prime}-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle=$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$

For the proof of Eq. (85), the proof will be similar to the proof of Eq.
(83)by reversing all inequalities and using Hansen-Pedersen’s inequality for
the operator monotone function $x^{p}$ for $0\leq p\leq 1$ which is

$\displaystyle
g_{\sigma}(\acute{\lambda}^{1-p}\mathcal{X}^{-1/2}\mathcal{A}_{i}^{p}\mathcal{X}^{-1/2})\preceq
g_{\sigma}((\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2})^{p})\preceq
g^{p}_{\sigma}(\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2}),$ (92)

where $1\leq i\leq k$. $\hfill\Box$

From the adjoint definition provided by Eq. (26), if we have Eq. (82), the
adjoint of $\mathfrak{M}$ will satsify

$\displaystyle\mathfrak{M}^{*}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{M}^{*}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}^{*}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(93)

where $p\geq 1$. Similarly, if we have Eq. (84), the adjoint of $\mathfrak{M}$
will satsify

$\displaystyle\mathfrak{M}^{*}(\mathcal{A}_{1}^{p},\cdots,\mathcal{A}_{k}^{p})$
$\displaystyle\succeq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{M}^{*}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}^{*}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(94)

where $0<p\leq 1$.

Next lemma is about validities of Eq. (83), Eq. (85), Eq. (93) and Eq. (94)
for special tensor means $\mathfrak{M}$.

###### Lemma 7

(1) Inequalities given by Eq. (83) and Eq. (85) are valid for
$\mathfrak{M}=\mathfrak{M}_{A,\bm{w}}$.
(2) Inequalities given by Eq. (93) and Eq. (94) are valid for
$\mathfrak{M}^{*}=\mathfrak{M}_{H,\bm{w}}$.

Proof: Let $\mathcal{A}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$ and
assume $1\leq p\leq 2$, by the operator convexity of $x^{p}$, we have

$\displaystyle\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}^{p}$
$\displaystyle\succeq$
$\displaystyle\left(\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}\right)^{p}$ (95)
$\displaystyle\succeq$
$\displaystyle\lambda_{\min}^{p-1}\left(\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}\right)\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}.$

From the induction argument used in Lemma 6, the inequality provided by Eq.
(95) can be extended to $p\geq 1$.

On the other hand, given $0<p\leq 1$, we have

$\displaystyle\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}^{p}$
$\displaystyle\preceq_{1}$
$\displaystyle\left(\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}\right)^{p}$ (96)
$\displaystyle\succeq_{2}$
$\displaystyle\lambda_{\min}^{p-1}\left(\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i}\right)\sum\limits_{i=1}^{k}w_{i}\mathcal{A}_{i},$

where $\preceq_{1}$ comes from the operator concavity of $x^{p}$, and
$\preceq_{2}$ comes from the following relation

$\displaystyle\mathcal{X}\succeq\lambda^{1-p}_{\min}(\mathcal{X})\mathcal{X}^{p},\leavevmode\nobreak\
\mbox{for $\mathcal{X}\in S_{\mbox{\tiny PD}}$}.$ (97)

From Eq. (95) and Eq. (96), the part (1) of this Lemma is proved. The part (2)
of this Lemma is also true since
$\mathfrak{M}_{H,\bm{w}}=\mathfrak{M}^{*}_{A,\bm{w}}$. $\hfill\Box$

We are ready to present the main theorem of this section.

###### Theorem 3

For $p\geq 1$, $0<q\leq 1$, $r\geq 1$, random tensors $\mathcal{X}_{i}\in
S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$ and a deterministic PD tensor
$\mathcal{C}$, we have

$\displaystyle\mathrm{Pr}\left(\lambda_{\min}^{p-1}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\npreceq\mathcal{C}\right)\leq\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right);$
(98)

Also, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})\npreceq\mathcal{C}\right)\leq$
(99)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

On the other hand, for $0<p\leq 1$, $0<q\leq 1$, random tensors
$\mathcal{X}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$ and a
deterministic PD tensor $\mathcal{C}$, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})\npreceq\mathcal{C}\right)\leq\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda^{p-1}_{\min}(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right);$
(100)

Also, we have

$\displaystyle\mathrm{Pr}\left(\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\npreceq\mathcal{C}\right)$
(101) $\displaystyle\leq$
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

Proof: Because the operation is $\\#_{q}$ is p.m.i., and we have
$\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})$ from the
solution of Eq. (30), and
$\mathfrak{P}_{\bm{w},-q}=\mathfrak{P}^{*}_{\bm{w},q}$ for $0<q\leq 1$, from
Lemma 6 and Lemma 7, we obtain the following:

$\displaystyle\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})$
$\displaystyle\succeq$
$\displaystyle\lambda_{\min}^{p-1}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$
$\displaystyle\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})$
$\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$
(102)

where $p\geq 1$.

For $0<p\leq 1$, we have

$\displaystyle\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})$
$\displaystyle\preceq$
$\displaystyle\lambda_{\min}^{p-1}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$
$\displaystyle\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})$
$\displaystyle\succeq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}).$
(103)

Apply Lemma 3 in [13] to above inequalities, we have desired results.
$\hfill\Box$

From Theorem 3, we can have the following Corollary about Karcher mean
$\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.

###### Corollary 1

Given $p\geq 1$, $r\geq 1$, random tensors $\mathcal{X}_{i}\in S_{\mbox{\tiny
PD}}$ for $1\leq i\leq k$ and a deterministic PD tensor $\mathcal{C}$, we have

$\displaystyle\mathrm{Pr}\left(\lambda^{p-1}_{\min}(\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\npreceq\mathcal{C}\right)\leq\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{G}_{\bm{w}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right);$
(104)

Also, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{G}_{\bm{w}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\npreceq\mathcal{C}\right)\leq$
(105)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda^{p-1}_{\max}(\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

On the other hand, given $0<p\leq 1$, $r\geq 1$, random tensors
$\mathcal{X}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$ and a
deterministic PD tensor $\mathcal{C}$, we have

$\displaystyle\mathrm{Pr}\left(\lambda^{p-1}_{\max}(\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\npreceq\mathcal{C}\right)\leq\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{G}_{\bm{w}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right);$
(106)

Also, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{G}_{\bm{w}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\npreceq\mathcal{C}\right)\leq$
(107)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda^{p-1}_{\min}(\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

Proof: As $q\searrow 0$, we have

$\displaystyle\lambda_{\min}(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\searrow\lambda_{\min}(\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})),$
(108)

and

$\displaystyle\lambda_{\max}(\mathfrak{P}_{\bm{w},-q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\nearrow\lambda_{\max}(\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})).$
(109)

By taking the limits of those inequalities in the proof in Theorem 3, we have

$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{G}_{\bm{w}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
(110) $\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$

where $p\geq 1$; and

$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{G}_{\bm{w}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
(111) $\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{G}_{\bm{w}}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$

where $0<p\leq 1$.

Apply Lemma 3 in [13] to above inequalities provided by Eq. (110) and (111),
we have desired results. $\hfill\Box$

##  6 Tail bounds for Multivariate Random Tensor Deformed Means Based on Ando-
Hiai Type Inequalities

In this section, we will generalize tail bounds for multivariate random tensor
for the deformed means $\mathfrak{M}_{\sigma_{1/p}}$ and
$\mathfrak{M}_{\sigma_{p}}$, where $\sigma_{1/p}$ and $\sigma_{p}$ are
operator means with respect to the representing functions
$g_{\sigma}(x^{1/p})$ and $g_{\sigma}(x^{p})$, respectively. We begin this
section with the following two lemmas. The first lemma is about
$\mathfrak{M}_{\sigma_{1/p}}$.

###### Lemma 8

Let $\mathfrak{M}:S^{k}_{\mbox{\tiny PD}}\rightarrow S_{\mbox{\tiny PD}}$ be a
map of _$k$ -variable tensor mean_ and $\sigma_{1/p}$ be the operator mean
with the representing function $g_{\sigma}(x^{1/p})$ for $p\geq 1$. Then

$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle\preceq_{1}$
$\displaystyle\mathfrak{M}_{\sigma_{1/p}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
(112) $\displaystyle\preceq_{2}$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$

Proof: We will prove the inequality $\preceq_{2}$ in Eq. (112) first. Let
$\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$, and
the statement (IV) in Theorem 1, it is enough for us to prove the following

$\displaystyle\lambda^{p-1}_{\max}(\mathcal{X})\mathcal{X}$
$\displaystyle\succeq$
$\displaystyle\mathfrak{M}((\lambda^{p-1}_{\max}(\mathcal{X})\mathcal{X})\sigma_{1/p}\mathcal{A}^{p}_{1},\cdots,(\lambda^{p-1}_{\max}(\mathcal{X})\mathcal{X})\sigma_{1/p}\mathcal{A}^{p}_{k}),$
(113)

which can also be expressed by

$\displaystyle\mathcal{I}$ $\displaystyle\succeq$
$\displaystyle\mathfrak{M}(g_{\sigma}(\lambda^{\frac{1}{p}-1}_{\max}(\mathcal{X})(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{1}\mathcal{X}^{-1/2})^{1/p}),\cdots,g_{\sigma}(\lambda^{\frac{1}{p}-1}_{\max}(\mathcal{X})(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{k}\mathcal{X}^{-1/2})^{1/p})).$
(114)

From Hansen-Pedersen’s inequality, for $1\leq i\leq k$, we have

$\displaystyle\lambda^{\frac{1}{p}-1}_{\max}(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{i}\mathcal{X}^{-1/2})^{1/p}$
$\displaystyle\preceq$
$\displaystyle\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2},$ (115)

then, from the monotonicity of $g_{\sigma}$ and the congruence property of
$\mathfrak{M}$, we have

$\displaystyle\mathfrak{M}(g_{\sigma}(\lambda^{\frac{1}{p}-1}_{\max}(\mathcal{X})(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{1}\mathcal{X}^{-1/2})^{1/p}),\cdots,g_{\sigma}(\lambda^{\frac{1}{p}-1}_{\max}(\mathcal{X})(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{k}\mathcal{X}^{-1/2})^{1/p}))$
(116) $\displaystyle\preceq$
$\displaystyle\mathfrak{M}(g_{\sigma}((\mathcal{X}^{-1/2}\mathcal{A}^{p}_{1}\mathcal{X}^{-1/2})^{1/p}),\cdots,g_{\sigma}((\mathcal{X}^{-1/2}\mathcal{A}^{p}_{k}\mathcal{X}^{-1/2})^{1/p}))=\mathcal{I}.$

This establishes the inequality $\preceq_{2}$ of Eq. (112).

The inequality $\preceq_{1}$ of Eq. (112) can be derived from $\preceq_{2}$ in
Eq. (112). By replacing $\mathfrak{M},\sigma$ and $\mathcal{A}_{i}$ for $1\leq
i\leq k$ in those terms in both sides of $\preceq_{2}$ and applying the
following relations:
$(\mathfrak{M}^{*})_{(\sigma^{*})_{1/p}}=(\mathfrak{M}_{\sigma_{1/p}})^{*}$
and $(\mathfrak{M}^{*})_{\sigma^{*}}=(\mathfrak{M}_{\sigma})^{*}$, we have

$\displaystyle\mathfrak{M}^{-1}_{\sigma_{1/p}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
$\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{M}^{-1}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}^{-1}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(117)

which is equivalent to

$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{M}_{\sigma_{1/p}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
(118)

Therefore, this lemma is proved. $\hfill\Box$

The next lemma is about $\mathfrak{M}_{\sigma_{p}}$.

###### Lemma 9

Let $\mathfrak{M}:S^{k}_{\mbox{\tiny PD}}\rightarrow S_{\mbox{\tiny PD}}$ be a
map of _$k$ -variable tensor mean_ and $\sigma_{p}$ be the operator mean with
the representing function $g_{\sigma}(x^{p})$ for $0<p\leq 1$. Then

$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{M}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle\preceq_{1}$
$\displaystyle\mathfrak{M}_{\sigma}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
(119) $\displaystyle\preceq_{2}$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$

Proof: We will prove the inequality $\preceq_{2}$ in Eq. (119) first. Let
$\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$, and
the statement (IV) in Theorem 1, it is enough for us to prove the following

$\displaystyle\lambda^{p-1}_{\min}(\mathcal{X})\mathcal{X}$
$\displaystyle\succeq$
$\displaystyle\mathfrak{M}((\lambda^{p-1}_{\min}(\mathcal{X})\mathcal{X})\sigma\mathcal{A}^{p}_{1},\cdots,(\lambda^{p-1}_{\min}(\mathcal{X})\mathcal{X})\sigma\mathcal{A}^{p}_{k}),$
(120)

which can also be expressed by

$\displaystyle\mathcal{I}$ $\displaystyle\succeq$
$\displaystyle\mathfrak{M}(g_{\sigma}(\lambda^{1-p}_{\min}(\mathcal{X})\mathcal{X}^{-1/2}\mathcal{A}^{p}_{1}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\lambda^{1-p}_{\min}(\mathcal{X})\mathcal{X}^{-1/2}\mathcal{A}^{p}_{k}\mathcal{X}^{-1/2})^{1/p}).$
(121)

From Hansen-Pedersen’s inequality, for $1\leq i\leq k$, we have

$\displaystyle\lambda^{1-p}_{\min}(\mathcal{X})\mathcal{X}^{-1/2}\mathcal{A}^{p}_{i}\mathcal{X}^{-1/2}$
$\displaystyle\preceq$
$\displaystyle(\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2})^{p},$
(122)

then, from the monotonicity of $g_{\sigma}$ and the congruence property of
$\mathfrak{M}$, we have

$\displaystyle\mathfrak{M}(g_{\sigma}(\lambda^{1-p}_{\min}(\mathcal{X})\mathcal{X}^{-1/2}\mathcal{A}^{p}_{1}\mathcal{X}^{-1/2}),\cdots,g_{\sigma}(\lambda^{1-p}_{\min}(\mathcal{X})\mathcal{X}^{-1/2}\mathcal{A}^{p}_{k}\mathcal{X}^{-1/2}))$
(123) $\displaystyle\preceq$
$\displaystyle\mathfrak{M}(g_{\sigma}((\mathcal{X}^{-1/2}\mathcal{A}_{1}\mathcal{X}^{-1/2})^{p}),\cdots,g_{\sigma}((\mathcal{X}^{-1/2}\mathcal{A}_{k}\mathcal{X}^{-1/2})^{p}))$
$\displaystyle=$
$\displaystyle\mathfrak{M}(g_{\sigma_{p}}(\mathcal{X}^{-1/2}\mathcal{A}_{1}\mathcal{X}^{-1/2}),\cdots,g_{\sigma_{p}}(\mathcal{X}^{-1/2}\mathcal{A}_{k}\mathcal{X}^{-1/2}))=\mathcal{I}.$

Therefore, we have proved the inequality $\preceq_{2}$ in Eq. (119).

The inequality $\preceq_{1}$ of Eq. (119) can be derived from $\preceq_{2}$ in
Eq. (119). By replacing $\mathfrak{M},\sigma$ and $\mathcal{A}_{i}$ for $1\leq
i\leq k$ in those terms in both sides of $\preceq_{2}$ and applying the
following relations:
$(\mathfrak{M}^{*})_{(\sigma^{*})_{p}}=(\mathfrak{M}_{\sigma_{p}})^{*}$ and
$(\mathfrak{M}^{*})_{\sigma^{*}}=(\mathfrak{M}_{\sigma})^{*}$, we have

$\displaystyle\mathfrak{M}^{-1}_{\sigma}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
$\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{M}^{-1}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}^{-1}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(124)

which is equivalent to

$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{M}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma_{p}}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{M}_{\sigma}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k}).$
(125)

Therefore, this lemma is proved. $\hfill\Box$

We are ready to present the main theorem of this section.

###### Theorem 4

For $p\geq 1$, $-1\leq q\leq 1$, $r\geq 1$, random tensors $\mathcal{X}_{i}\in
S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$ and a deterministic PD tensor
$\mathcal{C}$, we have

$\displaystyle\mathrm{Pr}\left(\lambda_{\min}^{p-1}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\npreceq\mathcal{C}\right)\leq$
(126)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{P}_{\bm{w},q/p}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right);$

Also, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{P}_{\bm{w},q/p}(\mathcal{X}_{1}^{p},\cdots,\mathcal{X}_{k}^{p})\npreceq\mathcal{C}\right)\leq$
(127)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

On the other hand, for $0<p\leq 1$, $-1\leq q\leq 1$, random tensors
$\mathcal{X}_{i}\in S_{\mbox{\tiny PD}}$ for $1\leq i\leq k$ and a
deterministic PD tensor $\mathcal{C}$, we have

$\displaystyle\mathrm{Pr}\left(\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\npreceq\mathcal{C}\right)$
(128) $\displaystyle\leq$
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left((\mathfrak{P}_{\bm{w},q}(\mathcal{X}^{p}_{1},\cdots,\mathcal{X}^{p}_{k}))\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right);$

Also, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{X}^{p}_{1},\cdots,\mathcal{X}^{p}_{k})\npreceq\mathcal{C}\right)$
(129) $\displaystyle\leq$
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda^{p-1}_{\min}(\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

Proof: From Lemma 8, we have

$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{P}_{\bm{w},q/p}(\mathcal{X}^{p}_{1},\cdots,\mathcal{X}^{p}_{k})$
(130) $\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},q}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$

where $p\geq 1$; and, from Lemma 9, we also have

$\displaystyle\lambda^{p-1}_{\max}(\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k})$
$\displaystyle\preceq$
$\displaystyle\mathfrak{P}_{\bm{w},q}(\mathcal{X}^{p}_{1},\cdots,\mathcal{X}^{p}_{k})$
(131) $\displaystyle\preceq$
$\displaystyle\lambda^{p-1}_{\min}(\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}))\mathfrak{P}_{\bm{w},qp}(\mathcal{X}_{1},\cdots,\mathcal{X}_{k}),$

where $0<p\leq 1$.

Apply Lemma 3 in [13] to above inequalities provided by Eq. (130) and (131),
we have desired results. $\hfill\Box$

##  7 Tail bounds for Multivariate Random Tensor Means Based on Reverse Ando-
Hiai Type Inequalities

Different from previous sections, we want to obtain the reverse of Ando-Hiai
type inequalities via Kantorovich constants. The Kantorovich constants are
defined by

$\displaystyle K(M,m,f,p)$ $\displaystyle=$
$\displaystyle\frac{mf(M)-Mf(m)}{(p-1)(M-m)}\left(\frac{(p-1)(f(M)-f(m))}{p(mF(M)-Mf(m))}\right)^{p},$
(132)

where $p$ is a real number and $f(t)$ is a real valued continuous function
defined on an real interval $[m,M]$ [26]. If $f(t)=t^{q}$, the Kantorovich
constants become

$\displaystyle K(M,m,t^{q},p)$ $\displaystyle=$
$\displaystyle\frac{mM^{q}-Mm^{q})}{(p-1)(M-m)}\left(\frac{(p-1)(M^{q}-m^{q})}{p(mM^{q}-Mm^{q})}\right)^{p}.$
(133)

Further, if we have $p=q$, we can express the Kantorovich constants as

$\displaystyle K(M,m,t^{p},p)$ $\displaystyle=$
$\displaystyle\frac{mM^{p}-Mm^{p})}{(p-1)(M-m)}\left(\frac{(p-1)(M^{p}-m^{p})}{p(mM^{p}-Mm^{p})}\right)^{p}$
(134) $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle K(M,m,p).$

Note that $K(M,m,p)>1$.

We have the following lemmas based on the Kantorovich constants.

###### Lemma 10

Let $\mathcal{A},\mathcal{B}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ be two PD tensors satisfying
$m\mathcal{I}\preceq\mathcal{A}\preceq M\mathcal{I}$, and
$\mathcal{B}^{2}\preceq\mathcal{I}$, where $0<m<M$. Given $p>1$, we have

$\displaystyle\mathcal{B}\star_{N}\mathcal{A}^{p}\star_{N}\mathcal{B}$
$\displaystyle\preceq$ $\displaystyle
K(M,m,p)(\mathcal{B}\star_{N}\mathcal{A}\star_{N}\mathcal{B})^{p},$ (135)

where $K(M,m,p)$ is defined by Eq (134).

Proof: From Corollary 2.12 in Book [26], we have

$\displaystyle\sum\limits_{j=1}^{k}\Phi_{j}(\mathcal{A}_{j}^{p})$
$\displaystyle\preceq$ $\displaystyle
K(M,m,p)\left(\sum\limits_{j=1}^{k}\omega_{j}\Phi_{j}(\mathcal{A}_{j})\right)^{p},$
(136)

where $\sum\limits_{j=1}^{k}\omega_{j}=1$. By setting $k=1$,
$\mathcal{A}_{j}=\mathcal{A}$, $\omega_{1}=1$, and
$\Phi(\mathcal{X})=\mathcal{B}\star_{N}\mathcal{X}\star_{N}\mathcal{B}$, where
$\mathcal{X}$ is any PD tensor, into Eq. (136), we have

$\displaystyle\mathcal{B}\star_{N}\mathcal{A}^{p}\star_{N}\mathcal{B}$
$\displaystyle\preceq$ $\displaystyle
K(M,m,p)(\mathcal{B}\star_{N}\mathcal{A}\star_{N}\mathcal{B})^{p}.$ (137)

Therefore, we obtain the desired inequality. $\hfill\Box$

###### Lemma 11

Let $\mathcal{A}_{i}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ be PD tensors satisfying
$m\mathcal{I}\preceq\mathcal{A}_{i}\preceq M\mathcal{I}$. Given $p>1$, we have

$\displaystyle\sum\limits_{i=1}^{k}\omega_{i}\mathcal{A}_{i}^{p}$
$\displaystyle\preceq$ $\displaystyle
K(M,m,p)\left(\sum\limits_{i=1}^{k}\omega_{i}\mathcal{A}_{i}\right)^{p}$ (138)

where $\omega_{i}\geq 0$ and $\sum\limits_{i=1}^{k}\omega_{i}=1$, i.e.
$(\omega_{1},\cdots,\omega_{k})$ is a probability vector.

Proof: From Theorem 3.18 in Book [26], we have

$\displaystyle\Phi(\mathcal{A}^{p})$ $\displaystyle\preceq$ $\displaystyle
K(M,m,p)\left(\Phi(\mathcal{A})\right)^{p},$ (139)

where $\Phi$ is a normalized positive linear map from a bounded linear
operator space over the Hilbert space $\mathrm{H}_{1}$, denoted by
$\mathfrak{B}(\mathrm{H}_{1})$ to another bounded linear operator space over
the Hilbert space $\mathrm{H}_{2}$, denoted by $\mathfrak{B}(\mathrm{H}_{2})$,
$p$ is greater or equal than $1$, and $\mathcal{A}$ is a PD tensor. Applying
Eq. (139) to the normalized positive linear map
$\Psi:\mathfrak{B}(\oplus_{1}^{k^{2}}\mathfrak{H})\rightarrow\mathfrak{B}(\mathfrak{H})$
defined by
$\Psi([\mathcal{A}_{i,j}]_{i,j=1}^{k})=\sum\limits_{i=1}^{k}\omega_{i}A_{i,i}$,
and assuming that $m\mathcal{I}\preceq A_{i,j}\preceq M\mathcal{I}$ for all
$1\leq i,j\leq k$, we have

$\displaystyle\sum\limits_{i=1}^{k}\omega_{i}\mathcal{A}_{i,i}^{p}$
$\displaystyle\preceq$ $\displaystyle
K(M,m,p)\left(\sum\limits_{i=1}^{k}\omega_{i}\mathcal{A}_{i,i}\right)^{p}.$
(140)

We prove this Lemma by replacing $\mathcal{A}_{i,i}$ with $\mathcal{A}_{i}$.
$\hfill\Box$

Following Theorem is about $k$-variable power means $\mathfrak{P}_{\bm{w},q}$
which can be treated as the reverse counterparts of Theorem 3.

###### Theorem 5

Let $\mathcal{A}_{i}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ be random PD tensors satisfying
$m\mathcal{I}\preceq\mathcal{A}_{i}\preceq M\mathcal{I}$ almost surely, a
deterministic PD tensor $\mathcal{C}\in\mathbb{C}^{I_{1}\times\dots\times
I_{N}\times I_{1}\times\dots\times I_{N}}$ and
$\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.
We also set $\lambda_{\min}(\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\lambda_{\min}$, and
$\lambda_{\max}(\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\lambda_{\max}$. Then, for any $q\in(0,1]$, $r\geq 1$ and $p\geq
1$, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{P}_{\bm{w},q}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\npreceq\mathcal{C}\right)\leq\mathrm{Tr}\left(\mathbb{E}\left[\left(\lambda_{\min}^{p-1}K_{1}K_{2}^{1/q}\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right),$
(141)

where $K_{1}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}K\left(M,m,p\right)$
and $K_{2}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}K\left(\frac{M^{q}}{\lambda^{q}_{\min}},\frac{m^{q}}{\lambda^{q}_{\max}},p\right)$.

On the other hand, for any $q\in[-1,0)$, $r\geq 1$ and $p\geq 1$, we have

$\displaystyle\mathrm{Pr}\left(\lambda_{\max}^{p-1}K^{-1}_{1}K^{\prime
1/q}_{2}\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\npreceq\mathcal{C}\right)\leq\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{P}_{\bm{w},q}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right),$
(142)

where $K^{\prime}_{2}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}K\left(\frac{\lambda^{q}_{\max}}{m^{q}},\frac{\lambda^{q}_{\min}}{M^{q}},p\right)$.

Proof: Since $\mathfrak{P}_{\bm{w},q}^{\mathrm{H}}=\mathfrak{P}_{\bm{w},-q}$,
$M^{-1}\mathcal{I}\preceq\mathcal{A}^{-1}_{i}\preceq m^{-1}\mathcal{I}$, we
will have Eq. (142) from Eq. (141) by replacing $\mathcal{A}_{i}$ with
$\mathcal{A}^{-1}_{i}$ and $q$ with $-q$. Therefore, it is enough to prove Eq.
(141) only.

Because
$\mathcal{X}=\sum\limits_{i=1}^{k}\omega_{i}\mathcal{X}\\#_{q}\mathcal{A}_{i}$,
we have

$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{k}\omega_{i}\left(\mathcal{X}^{-1/2}\star_{N}\mathcal{A}_{i}\star_{N}\mathcal{X}^{-1/2}\right)^{q};$
(143)

and the fact that $\lambda_{\min}\mathcal{X}^{-1}\preceq\mathcal{I}$, we have

$\displaystyle(\lambda^{1/2}_{\min}\mathcal{X}^{-1/2})\star_{N}\mathcal{A}_{i}^{p}\star_{N}(\lambda^{1/2}_{\min}\mathcal{X}^{-1/2})\preceq$
(144) $\displaystyle
K(M,m,p)\left[(\lambda^{1/2}_{\min}\mathcal{X}^{-1/2})\star_{N}\mathcal{A}_{i}\star_{N}(\lambda^{1/2}_{\min}\mathcal{X}^{-1/2})\right]^{p},$

by Lemma 10. Since
$\lambda_{\min}(\mathcal{X})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\lambda_{\min}$, and
$K\left(M,m,p\right)\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}K_{1}$, we
have

$\displaystyle\sum\limits_{i=1}^{k}\omega_{i}\left[\lambda_{\min}^{1-p}K^{-1}_{1}(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{i}\mathcal{X}^{-1/2})\right]^{q}\preceq\sum\limits_{i=1}^{k}\omega_{i}(\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2})^{qp}.$
(145)

Besides, due to
$\frac{m}{\lambda_{\max}}\mathcal{I}\preceq\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2}\preceq\frac{M}{\lambda_{\min}}\mathcal{I}$
and $0<q\leq 1$, we also have

$\displaystyle\left(\frac{m}{\lambda_{\max}}\right)^{q}\mathcal{I}\preceq\left(\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2}\right)^{q}\preceq\left(\frac{M}{\lambda_{\min}}\right)^{q}\mathcal{I}.$
(146)

From Lemma 11 and Eq. (145), we have

$\displaystyle\sum\limits_{i=1}^{k}\omega_{i}\left[\lambda_{\min}^{1-p}K^{-1}_{1}(\mathcal{X}^{-1/2}\mathcal{A}^{p}_{i}\mathcal{X}^{-1/2})\right]^{q}\preceq$
(147) $\displaystyle
K\left(\frac{M^{q}}{\lambda^{q}_{\min}},\frac{m^{q}}{\lambda^{q}_{\max}},p\right)\left[\sum\limits_{i=1}^{k}\omega_{i}\left(\mathcal{X}^{-1/2}\mathcal{A}_{i}\mathcal{X}^{-1/2}\right)^{q}\right]^{p}.$

By setting
$K\left(\frac{M^{q}}{\lambda^{q}_{\min}},\frac{m^{q}}{\lambda^{q}_{\max}},p\right)\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}K_{2}$ and Eq. (147), we have

$\displaystyle\sum\limits_{i=1}^{k}\omega_{i}\left[\left(\lambda_{\min}^{p-1}K_{1}K_{2}^{1/q}\mathcal{X}\right)\\#_{q}\mathcal{A}_{i}^{p}\right]\preceq\lambda_{\min}^{p-1}K_{1}K_{2}^{1/q}\mathcal{X}.$
(148)

By Theorem 1, we have

$\displaystyle\mathfrak{P}_{\bm{w},q}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\preceq\lambda_{\min}^{p-1}K_{1}K_{2}^{1/q}\mathfrak{P}_{\bm{w},q}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}),$
(149)

then, we apply Lemma 3 in [13] to Eq. (149) to obtain Eq. (141). $\hfill\Box$

Following Theorem is about the deformed mean $\mathfrak{M}_{\sigma_{1/p}}$ and
this theorem can be treated as the reverse counterparts of Lemma 8.

###### Theorem 6

Let $\mathcal{A}_{i}\in\mathbb{C}^{I_{1}\times\dots\times I_{N}\times
I_{1}\times\dots\times I_{N}}$ be random PD tensors satisfying
$m\mathcal{I}\preceq\mathcal{A}_{i}\preceq M\mathcal{I}$ almost surely, a
deterministic PD tensor $\mathcal{C}\in\mathbb{C}^{I_{1}\times\dots\times
I_{N}\times I_{1}\times\dots\times I_{N}}$ and
$\mathcal{X}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$.
Then, for any $p\geq 1$ and $r\geq 1$, we have

$\displaystyle\mathrm{Pr}\left(\mathfrak{M}_{\sigma_{1/p}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\npreceq\mathcal{C}\right)\leq$
(150)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(K_{1}\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right),$

where $K_{1}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}K(M,m,p)$. On the
other hand, we also have

$\displaystyle\mathrm{Pr}\left(K_{1}^{-1}\lambda^{p-1}_{\max}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})\npreceq\mathcal{C}\right)\leq$
(151)
$\displaystyle\mathrm{Tr}\left(\mathbb{E}\left[\left(\mathfrak{M}_{\sigma_{1/p}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})\right)^{r}\right]\star_{N}\mathcal{C}^{-1}\right).$

Proof: Because of
$\lambda_{\min}(\mathcal{X})\mathcal{X}^{-1}\preceq\mathcal{I}$ and Lemma 10,
we have

$\displaystyle(\lambda^{1/2}_{\min}(\mathcal{X})\mathcal{X}^{-1/2})\star_{N}\mathcal{A}_{i}^{p}\star_{N}(\lambda^{1/2}_{\min}(\mathcal{X})\mathcal{X}^{-1/2})\preceq$
(152) $\displaystyle
K(M,m,p)\left[(\lambda^{1/2}_{\min}(\mathcal{X})\mathcal{X}^{-1/2})\star_{N}\mathcal{A}_{i}\star_{N}(\lambda^{1/2}_{\min}(\mathcal{X})\mathcal{X}^{-1/2})\right]^{p}.$

Similar to the proof in Lemma 8, we will obtain

$\displaystyle
K^{-1}(M,m,p)\lambda^{p-1}_{\max}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k})$
(153) $\displaystyle\preceq$
$\displaystyle\mathfrak{M}_{\sigma_{1/p}}(\mathcal{A}^{p}_{1},\cdots,\mathcal{A}^{p}_{k})$
$\displaystyle\preceq$ $\displaystyle
K(M,m,p)\lambda^{p-1}_{\min}(\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}))\mathfrak{M}_{\sigma}(\mathcal{A}_{1},\cdots,\mathcal{A}_{k}).$

Then, we can apply Lemma 3 in [13] to both inequalities Eq. (153) to obtain
results of this theorem. $\hfill\Box$

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