Order Relations of the Wasserstein mean and the spectral geometric mean

The primary objective of this manuscript is to investigate the near order and the Löwner order relations concerning the Wasserstein mean and the spectral geometric mean, as well as the curves induced by these two means.

Let ${\mathbb{C}}_{n\times n}$ be the space of all $n\times n$ complex matrices. In ${\mathbb{C}}_{n\times n}$, let $\mathbb{H}_{n}$ (resp. $\mathbb{P}_{n}$, $\overline{\mathbb{P}}_{n}$) be the set of $n\times n$ Hermitian (resp. positive definite, positive semidefinite) matrices, and $\mathrm{U}(n)$ the group of $n\times n$ unitary matrices. Given $A\in{\mathbb{C}}_{n\times n}$, we denote $|A|=(A^{*}A)^{1/2}$. For $A\in\mathbb{P}_{n}$, let $\lambda(A)$ denote the $n$-tuple of eigenvalues of $A$ with nonincreasing order, that is, $\lambda(A)=(\lambda_{1}(A),\lambda_{2}(A),\dots,\lambda_{n}(A))$ and $\lambda_{1}(A)\geq\lambda_{2}(A)\geq\cdots\geq\lambda_{n}(A)$.

On the space of positive definite matrices, several operator means are popular and have been studied extensively. Given $A,B\in\mathbb{P}_{n}$, the arithmetic mean, the Wasserstein mean, the metric geometric mean, and the spectral geometric mean are defined for $t\in[0,1]$:

where $A\sharp B=A\sharp_{1/2}B$. Many discovered relations between these means are related to their spectra.

The metric geometric mean was first introduced by Pusz and Woronowicz [24] for $t=1/2$ and extended to $t\in[0,1]$ by Kubo and Ando [20]. The mean was extended to multiple variables by Ando, Li and Mathias [5]. The properties of this mean were studied extensively by Lim [23]. The spectral geometric mean was introduced by Fiedler and Pták [10] for $t=1/2$ and extended to $t\in[0,1]$ by Lee and Lim [22]. Properties of spectral geometric mean can be found in [2, 11, 13, 17, 18] and the references therein. The Wasserstein mean is linked to the barycenter in the Wasserstein space of Gaussian distributions, which is one of the popular topics in matrix analysis and probability theory [1, 3]. Many interesting inequalities and properties of the Wasserstein mean has been given in [8, 11, 14, 15]. Hwang and Kim in [16, Lemma 2.4] gave another form of the Wasserstein mean (1.2) in terms of the metric geometric mean and arithmetic mean as

This form is analogous to the form of the spectral geometric mean (1.4). In consequence, many properties of these two means are similar.

The following interesting relations between positive definite matrices will be discussed.

1. (1)

   We use $A\geq 0$ to denote that $A$ is positive semidefinite. Two matrices $A,B\in\mathbb{P}_{n}$ satisfy the Löwner order $A\leq B$ if $B-A\geq 0$.

2. (2)

   Define the near order on $\mathbb{P}_{n}$: $A\preceq B$ if $A^{-1}\sharp B\geq I$. The relation is introduced by Dumitru and Franco [9].

3. (3)

   Given $A,B\in\mathbb{P}_{n}$, we define the eigenvalue entrywise order relation: $A\leq_{\lambda}B$ if $\lambda_{i}(A)\leq\lambda_{i}(B)$ for $1\leq i\leq n$. We say that $A=_{\lambda}B$ if $\lambda(A)=\lambda(B)$.

4. (4)

   Given $A,B\in\mathbb{P}_{n}$, we write $A\prec_{w\log}B$ if $\lambda(A)$ is weakly log-majorized by $\lambda(B)$, that is,

   $$\prod_{i=1}^{k}\lambda_{i}(A)\leq\prod_{i=1}^{k}\lambda_{i}(B),\quad k=1,2,\dots,m.$$

   (1.6)

   In particular, we say that $\lambda(A)$ is log-majorized by $\lambda(B)$, denoted by $A\prec_{\log}B$, if (1.6) is true for $k=1,2,\dots,n-1$ and equality holds for $k=m$.

The above relations satisfy the transitive property except for the near order. The Löwner order is the strongest condition among these relations. The near order is weaker than the Löwner order but stronger than the eigenvalue entrywise order, which we will prove in Theorem 2.4. It is straightforward that $A\leq_{\lambda}B$ implies $A\prec_{w\log}B$. So for $A,B\in\mathbb{P}_{n}$, we have the following relationships:

In Theorem 3.1, we show that $A\natural_{t}B\preceq A\diamond_{t}B$ for $t\in(0,1)$, which completes a chain of order relations among means in (2.3). Consequently, there is the eigenvalue entrywise relation $A\natural_{t}B\leq_{\lambda}A\diamond_{t}B$ for $t\in(0,1)$.

The metric geometric mean satisfies that if $A\leq B$, then the induced geodesic curve $\{A\sharp_{t}B\mid t\geq 0\}$ is monotonically increasing in terms of the Löwner order $\leq$ with respect to $t$. Similar properties hold for the curves induced by the Euclidean mean $A\nabla_{t}B$ and the log-Euclidean mean $\exp({(1-t)\log A+t\log B})$, but not for the curves induced by the Wasserstein mean and the spectral geometric mean. We show in Theorems 3.6 that if $A\preceq B,$ then $\{A\diamond_{t}B\mid t\geq 0\}$ and $\{A\natural_{t}B\mid t\in\mathbb{R}\}$ are monotonically increasing in terms of the near order $\preceq$ with respect to $t$. In Theorem 3.7, the near order relations between $A\diamond_{t}B$ and $A\diamond_{s}B$, between $A\natural_{t}B$ and $A\natural_{s}B$, and between $A\natural_{t}B$ and $A\diamond_{s}B$, are compared, respectively. For certain real powers $p$, the near order on the curves $\{A^{p}\diamond_{t}B^{p}\mid t\geq 0\}$ and $\{B^{-p}\diamond_{t}A^{-p}\mid t\geq 0\}$ are also discussed. The results disclose the existence of abundant near order relations and the corresponding eigenvalue entrywise order relations in the Wasserstein mean, the spectral geometric mean, and the curves induced by two means.

In Section 4, we study the Löwner order properties of Wasserstein mean and spectral geometric mean, both of which are connected to the metric geometric mean. In particular, Theorem 4.2 shows that if $A\diamond_{t}B\leq A\diamond_{t}C$ or $A\natural_{t}B\leq A\natural_{t}C$ for one $t\in(0,1]$, then $A^{-1}\sharp_{s}B\leq A^{-1}\sharp_{s}C$ for all $s\in[0,1/2]$.

Some preliminary results of the metric geometric mean, the Wasserstein mean, the spectral geometric mean, and their relations are explored in Section 2.

Extensive investigations have been done on the properties of matrix means. Here we list some basic properties of the metric geometric mean [7, 20, 22], the spectral geometric mean [22, 23] and the Wasserstein mean [14, 16, 19]. They display different characteristics of these three means.

Besides the individual properties of matrix means, many relations among these means have been discovered. The near order relation $\preceq$ on $\mathbb{P}_{n}$ is recently introduced by Dumitru and Franco [9]:

$A\preceq B$ if and only if $A^{-1}\sharp B\geq I$.

It is natural that $A\preceq B$ if and only if $B^{-1}\preceq A^{-1}$. However, the relation “$\preceq$” does not satisfy transitive property [9], that is, $A\preceq B$ and $B\preceq C$ do not necessarily imply that $A\preceq C$. So the relation “$\preceq$” is called a near order.

On one hand, the near order relation is weaker than the Löwner order, since $A\leq B$ implies that $A^{-1}\sharp B\geq A^{-1}\sharp A=I$ so that $A\preceq B$. On the other hand, we show below that the near order is stronger than the eigenvalue entrywise relation.

From [4, 6, 12], we have the log-majorization relations between the metric geometric mean, the log-Euclidean mean, the Fidelity mean and the spectral geometric mean as

It is well-known (e.g. [21, 8] ) that there are Löwner orders:

Our new result in Theorem 3.1 shows that $A\natural_{t}B\preceq A\diamond_{t}B$ for all $t\in(0,1)$. It completes a chain of order relations among means as

The relation between any two of the above means can be derived from (2.3) and the transitive properties. Here are some relations:

1. (1)

   (2.3) and [9, Theorem 2] imply that

   $$A\natural_{t}B\preceq A\nabla_{t}B.$$

   (2.4)

2. (2)

   From (2.3), we get

   $$A\sharp_{t}B\prec_{w\log}A\diamond_{t}B.$$

   (2.5)

   It is the known strongest relation. The following counterexample shows that $A\sharp_{t}B$ and $A\diamond_{t}B$ do not have eigenvalue entrywise relation, near order relation, or Löwner order relation. Let

   $$A=\begin{bmatrix}39.1195&42.1116\\ 42.1116&61.1568\end{bmatrix}\quad\text{and}\quad B=\begin{bmatrix}26.3279&13.3485\\ 13.3485&12.2727\end{bmatrix}.$$

   Then we have

   $$A\sharp_{t}B=\begin{bmatrix}32.2446&29.2497\\ 29.2497&39.8872\end{bmatrix}\quad\text{and}\quad A\diamond_{t}B=\begin{bmatrix}35.6339&33.9111\\ 33.9111&45.3815\end{bmatrix}.$$

   The spectra of $A\sharp_{t}B$ is $\{65.5641,6.5677\}$ and that of $A\diamond_{t}B$ is $\{74.7672,6.2481\}$. Thus $A\sharp_{t}B$ and $A\diamond_{t}B$ do not have eigenvalue entrywise relation. (1.7) implies that $A\sharp_{t}B$ and $A\diamond_{t}B$ do not have either near order relation or Löwner order relation.

3. (3)

   (2.3) may not always provide the strongest relation between means. For example, (2.3) implies that $(A^{-1}\nabla_{t}B^{-1})^{-1}\prec_{w\log}A\diamond_{t}B$. However, Theorem 2.3(6) and Theorem 3.9 show that

   $$(A^{-1}\nabla_{t}B^{-1})^{-1}\leq(A^{-1}\diamond_{t}B^{-1})^{-1}\preceq A\diamond_{t}B\leq A\nabla_{t}B,$$

   (2.6)

   hence by [9, Theorem 2], a stronger relation exists:

   $$(A^{-1}\nabla_{t}B^{-1})^{-1}\preceq A\diamond_{t}B.$$

   (2.7)

A relation between the near order and the Löwner order is given as follows, which strengthen a result in [16, Hwang and Kim, 2022].

By [26, Theorem II.1], on the manifold $\overline{P_{n}}$ of $n\times n$ positive semidefinite matrices, the formula (1.2) of $A\diamond_{t}B$ can be extended to all $t\in\mathbb{R}$, and

in which $U$ is a unitary matrix in the polar decomposition $B^{1/2}A^{1/2}=U|B^{1/2}A^{1/2}|$, and $\nabla_{t}$ in (1.1) is extended to all $t\in\mathbb{R}.$ When $A\in\mathbb{P}_{n},$ (2.8) is equivalent to that for $t\in\mathbb{R}$:

Likewise, the formula (1.4) of $A\natural_{t}B$ can be extended to all $t\in\mathbb{R}$ such that

(2.9) and (2.10) immediately imply the following result.

All properties of the spectral geometric mean in Theorem 2.2 still hold for $s,u,t\in\mathbb{R}$, and they can be proved by (2.10). The extension of Theorem 2.2(4) to $s,u,t\in\mathbb{R}$ is proved below.

However, the counterpart of Proposition 2.8 for $A\diamond_{t}B$ is not true, since the Wasserstein curve $\{A\diamond_{t}B\mid t\in\mathbb{R}\}$ given by (1.2), (2.8), or (2.9) consists of several geodesic curves joint at some boundary points of $\overline{P_{n}}.$ A difference between $\natural_{t}$ and $\diamond_{t}$ lies in the fact that: in (2.10), $(A^{-1}\sharp B)^{t}$ for $t\in\mathbb{R}$ is always positive definite, but in (2.9), $I\nabla_{t}(A^{-1}\sharp B)$ is not.

When $A$ and $B$ has near order relation, the following properties hold for Wasserstein curve.

In the coming sections, we will focus on exploring the near order relation and the Löwner order relations between the spectral geometric mean and the Wasserstein mean, and these relations on the curves defined by two means.

There are abundant near order relations on the curves originated from the Wasserstein mean and the spectral geometric mean. We will explore these relations here.

$A\diamond_{t}B$ and $A\natural_{t}B$ have the following near order relations.

Theorem 2.4 shows that the near order is stronger than the eigenvalue entrywise relation. So (3.1) implies the following result.

Corollary 3.3 strengthens the weak log-majorization result in [11]: for $t\in(0,1)$, $A\natural_{t}B\prec_{w\log}A\diamond_{t}B$.

Theorem 2.3(5) shows that for $t\in(0,1),$ we have $\det(A\diamond_{t}B)\geq(\det A)^{1-t}(\det B)^{t}=\det(A\natural_{t}B)$. The following is a direct consequence of Theorem 3.1.

Next, we study the monotonicity of near order on the curves defined by $\diamond$ and $\natural$.