Affine-invariant midrange statistics^††thanks: This work has received support from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645. Cyrus Mostajeran is supported by the Cambridge Philosophical Society.

Consider the family of affine-invariant metric distances $d_{\Phi}$ on $\mathbb{P}(n)$ defined as

where $\|\cdot\|_{\Phi}$ is any unitarily invariant norm on the space of Hermitian matrices of dimension $n$ defined by $\|X\|_{\Phi}:=\Phi(\lambda_{1}(X),\ldots,\lambda_{n}(X)),$ with $\lambda_{\min}(X)=\lambda_{n}(X)\leq\ldots\leq\lambda_{1}(X)=\lambda_{\max}(X)$ denoting the $n$ real eigenvalues of $X$ and $\Phi$ a symmetric gauge norm on $\mathbb{R}^{n}$ (i.e. a norm that is invariant under permutations and sign changes of the coordinates) [1]. For any such $\Phi$, $X$ is said to be a $d_{\Phi}$-midpoint of $A$ and $B$ if

The curve $\gamma_{\mathcal{G}}:[0,1]\rightarrow\mathbb{P}(n)$ defined by

is geometrically significant as a minimal geodesic for any of the affine-invariant metrics $d_{\Phi}$ [1]. The midpoint of $\gamma_{\mathcal{G}}$ is the matrix geometric mean $A\#B$, which is a metric midpoint in the sense of $d_{\Phi}(A,A\#B)=d_{\Phi}(A\#B,B)=\frac{1}{2}d_{\Phi}(A,B)$ for any $\Phi$. For the choice of $\Phi(x_{1},\ldots,x_{n})=(\sum_{i}x_{i}^{2})^{1/2}$, $d_{2}:=d_{\Phi}$ corresponds to the metric distance generated by the standard affine-invariant Riemannian structure given by $\langle X,Y\rangle_{\Sigma}=\operatorname{tr}(\Sigma^{-1}X\Sigma^{-1}Y)$ for $\Sigma\in\mathbb{P}(n)$ and Hermitian matrices $X,Y\in T_{\Sigma}\mathbb{P}(n)$. For the choice of $\Phi(x_{1},\ldots,x_{n})=\max_{i}|x_{i}|$ on the other hand, $d_{\infty}:=d_{\Phi}$ yields the distance function that coincides with the Thompson metric [7] on the cone $\mathbb{P}(n)$

While the minimal geodesic $\gamma_{\mathcal{G}}$ in (3) and the midpoint is unique for the Riemannian distance function $d_{2}$, it is not unique with respect to the $d_{\infty}$ metric which generally admits infinitely many minimal geodesics and midpoints between a given pair of matrices $A,B\in\mathbb{P}(n)$ [6], as expected from the analogous picture concerning the associated norms in $\mathbb{R}^{n}$. As we shall see in Section 2, some of these minimal geodesics are much more readily constructible than others from a computational standpoint and yield midpoints that satisfy many of the properties expected of an affine-invariant matrix mean. Specifically, we will use the midpoint of a particular minimal geodesic of $d_{\infty}$ from a construction by Nussbaum as a scalable relaxation of the matrix geometric mean that is much cheaper to construct than $A\#B$.

Given a collection of $N$ points $Y_{i}$ in $\mathbb{P}(n)$, the midrange problem can be formulated as the following optimization problem

We call a solution $X^{\star}$ to the above problem a midrange of $\{Y_{i}\}$.

As $\gamma_{\mathcal{G}}$ is a minimal geodesic for the $d_{\infty}$ metric, $A\#B$ lies in the set of midpoints of this metric. In particular,

Recall that the metric distance $d_{\infty}$ coincides with the Thompson metric $d_{T}$ on the cone $\mathbb{P}(n)$. If $C$ is a closed, solid, pointed, convex cone in a vector space $V$, then $C$ induces a partial order on $V$ given by $x\leq y$ if and only if $y-x\in C$. The Thompson metric on $C$ is defined as $d_{T}(x,y)=\log\max\{M(x/y;C),M(y/x;C)\}$, where $M(y/x;C):=\inf\{\lambda\in\mathbb{R}:y\leq\lambda x\}$ for $x\in C\setminus\{0\}$ and $y\in V$. For $A,B\in\mathbb{P}(n)$, we have $M(A/B)=\lambda_{\max}(AB^{-1})$, so that

which indeed coincides with $d_{\infty}$ in (4).

It is known that the Thompson metric does not admit unique minimal geodesics. Indeed, a remarkable construction by Nussbaum describes a family of geodesics that generally consists of an infinite number of curves connecting a pair of points in a cone $C$. In particular, setting $\alpha:=1/M(x/y;C)$ and $\beta:=M(y/x;C)$, the curve $\phi:[0,1]\rightarrow C$ given by

is always a minimal geodesic from $x$ to $y$ with respect to the Thompson metric. The curve $\phi$ defines a projective straight line in the cone [6]. If we take $C$ to be the cone of positive semidefinite matrices with interior $\operatorname{int}C=\mathbb{P}(n)$, then for a pair of points $A,B\in\mathbb{P}(n)$, we have $\beta=M(B/A;C)=\lambda_{\max}(BA^{-1})$ and $\alpha=1/M(A/B;C)=\lambda_{\min}(BA^{-1})$. Thus, the minimal geodesic described by (10) takes the form

where $\lambda_{\max}$ and $\lambda_{\min}$ denote the largest and smallest eigenvalues of $BA^{-1}$, respectively. Taking the midpoint $t=1/2$ of this geodesic, we arrive at a computationally convenient $d_{\infty}$-midpoint of $A$ and $B$, which we will denote by $A*B$.

The result follows from elementary algebraic simplification upon setting $A*B=\phi(1/2;A,B)$ in the case $\lambda_{\min}\neq\lambda_{\max}$. If $\lambda_{\min}=\lambda_{\max}$, then $\phi(1/2;A,B)=\sqrt{\lambda_{\min}}A$ also agrees with the formula in (12). The geometry of the set of $d_{\infty}$-midpoints of a given pair of points in $\mathbb{P}(n)$ is studied in detail in [5], where it is shown that there is a unique $d_{\infty}$ minimal geodesic from $A$ to $B$ if and only if the spectrum of $BA^{-1}$ lies in a set $\{\lambda,\lambda^{-1}\}$ for some $\lambda>0$. Moreover, it is shown that otherwise there are infinitely many $d_{\infty}$ minimal geodesics from $A$ to $B$, and that the set of $d_{\infty}$-midpoints of $A$ and $B$ is compact and convex in both Riemannian and Euclidean senses [5].

We now consider the merits of $A*B$ as a mean of $A$ and $B$. The following are a number of properties that should be satisfied by a sensible notion of a mean $\mu:\mathbb{P}(n)\times\mathbb{P}(n)\rightarrow\mathbb{P}(n)$ of a pair of positive definite matrices [2, 4].

1. (i)

   Continuity: $\mu$ is a continuous map.

2. (ii)

   Symmetry: $\mu(A,B)=\mu(B,A)$ for all $A,B\in\mathbb{P}(n)$.

3. (iii)

   Affine-invariance: $\mu(XAX^{*},XBX^{*})=X\mu(A,B)X^{*}$, for all $X\in GL(n)$.

4. (iv)

   Order property: $A\preceq B\implies A\preceq\mu(A,B)\preceq B$.

5. (v)

   Monotonicity: $\mu(A,B)$ is monotone in its arguments.

Note that $X^{*}$ in (iii) denotes the conjugate transpose of $X$. It is relatively straightforward to show that the map $\mu(A,B):=A*B$ satisfies properties (i)-(iii) listed above. In the remainder of this section, we will turn our attention to the order and monotonicity properties (iv) and (v).

Condition (iv) is a generalization of the property of means of positive numbers whereby a mean of a pair of points is expected to lie between the two points on the number line. For Hermitian matrices, a standard partial order $\preceq$ exists according to which $A\preceq B$ if and only if $B-A$ is positive semidefinite. This partial order is known as the Löwner order and the monotonicity of condition (v) is also with reference to this order. It is well-known that the Löwner order is affine-invariant in the sense that for all $A,B\in\mathbb{P}(n)$, $X\in GL(n)$, $A\preceq B$ implies that $XAX^{*}\preceq XBX^{*}$. In particular, $A\preceq B$ if and only if $I\preceq A^{-1/2}BA^{-1/2}$. Thus, by affine-invariance of $\mu$, it suffices to prove (iv) in the case where $A=I$ since $A\preceq\mu(A,B)\preceq B$ if and only if $I\preceq\mu(I,A^{-1/2}BA^{-1/2})\preceq A^{-1/2}BA^{-1/2}$. To establish condition (iv) for $\mu(A,B)=A*B$, we shall make use of the following lemma whose proof is elementary.

Let $\Sigma\in\mathbb{P}(n)$ be such that $I\preceq\Sigma$ and note that this is equivalent to $\lambda_{i}(\Sigma)\geq 1$ for $i=1,\ldots,n$. Writing $\lambda_{\min}=\lambda_{\min}(\Sigma)$ and $\lambda_{\max}=\lambda_{\max}(\Sigma)$, we have by Lemma 1 that

since $\lambda_{i}(\Sigma)\geq\lambda_{\min}\geq 1$. Thus, we have shown that $I\preceq\Sigma$ implies $I\preceq I*\Sigma$. To prove the other inequality, note that

Therefore, we have also shown that $\Sigma-I*\Sigma\succeq 0$. That is,

for all $\Sigma\in\mathbb{P}(n)$. In particular, upon substituting $\Sigma=A^{-1/2}BA^{-1/2}$ in (20) and using the affine-invariance properties of both the Löwner order and the mean $\mu(A,B)=A*B$, we establish the following important property.

We now consider the monotonicity of $\mu$ in its arguments. First recall that a map $F:\mathbb{P}(n)\rightarrow\mathbb{P}(n)$ is said to be monotone if $\Sigma_{1}\preceq\Sigma_{2}$ implies that $F(\Sigma_{1})\preceq F(\Sigma_{2})$. By symmetry and affine-invariance, it is sufficient to consider monotonicity of $\mu(I,\Sigma)$ with respect to $\Sigma$. That is, monotonicity is established by showing that

Unfortunately, it turns out that $F(\Sigma):=I*\Sigma$ is not monotone with respect to $\Sigma$ as we shall demonstrate below. Nonetheless, $F$ is seen to enjoy certain weaker monotonicity properties, which is interesting and insightful. Considering the eigenvalues of $I*\Sigma$, we find that

where $\lambda_{\min}$ and $\lambda_{\max}$ refer to the smallest and largest eigenvalues of $\Sigma$.

It is in the sense of the above that $\mu(A,B)=A*B$ inherits a weak monotonicity property. The monotonic dependence of the extremal eigenvalues of $I*\Sigma$ on $\Sigma$ ensures that if $\Sigma_{1}\preceq\Sigma_{2}$, then we can at least rule out the possibility that $I*\Sigma_{1}\succ I*\Sigma_{2}$, where $\succ 0$ here denotes positive definiteness. To prove that monotonicity is generally not satisfied in the full sense of condition (v), consider a diagonal matrix $\Sigma=\operatorname{diag}(a,b,x)\in\mathbb{P}(3)$, where $\lambda_{\min}(\Sigma)=a<b\leq x=\lambda_{\max}(\Sigma)$ and $x$ is thought of as a variable. We have $I*\Sigma=\operatorname{diag}\left(\sqrt{a},f(x),\sqrt{x}\right)$, where

Taking the derivative of $f$ with respect to $x$, we find that

which shows that the second eigenvalue of $I*\Sigma$ decreases as $x$ increases. Thus, we see that $I*\Sigma$ cannot depend monotonically on $\Sigma$ in this example.

Before completing this section on the midrange $\mu(A,B)=A*B$ of a pair of positive definite matrices, we note a key scaling property satisfied by $\mu$ which suggests that it is a plausible candidate as a scalable substitute for the standard matrix geometric mean $A\#B$.

The scaling in (26) of course does not generally hold for a mean of two matrices. Indeed, it does not generally hold for means arising as $d_{\infty}$-midpoints either. For instance, [5] identifies

as another $d_{\infty}$-midpoint of $A$ and $B$ corresponding to the midpoint of a different $d_{\infty}$ minimal geodesic to the one considered in this paper. It is clear that (27) does not scale geometrically as in (26). As a summary, we collect the key results of this section in the following theorem.