REMARKS ON CONVERGENCE OF INDUCTIVE MEANS

Abstract

We define new inductive mean constructed by a mean on a complete metric space, and see its convergence when the intrinsic mean is given. We also give many examples of inductive matrix means and claim that the limit of inductive mean constructed by the intrinsic mean is not the Karcher mean, in general.

1. Introduction

Among a variety of applications for positive definite matrices, the process of averaging has become attractive and widely studied. Since the two-variable geometric mean of positive definite matrices A and B

has been introduced by Kubo and Ando [9], its algebraic and geometric properties have been studied. On the open convex cone of positive definite matrices equipped with the Riemannian trace metric d(X, Y) = ∥ log(X−1/2Y X−1/2)∥2, the unique geodesic connecting from A to B is given by

See [3] for more details. An extensive theory of two-variable geometric mean to the multivariable geometric mean has sprung up and has remained problematic. Ando, Li, and Mathias [1] have especially suggested a symmetrization procedure to the multivariable geometric mean of positive definite matrices including ten desirable properties for extended geometric means. Moreover, a convergence for symmetrization procedure has been recently proved by Kim and Petz [8].

A natural and attractive average of positive definite matrices is the least squares mean, also called the Cartan mean, Riemannian barycenter, and Karcher mean. The Karcher mean ∧ (ω;A1, . . . ,An) of positive definite matrices A1, . . . ,An and a positive probability vector ω = (w1, . . . ,wn) is defined as the unique minimizer (provided it exists) of the weighted sum of squares of the Riemannian trace distances to each of the point. That is,

In general, such minimizer exists uniquely on a Hadamard space, which is a complete metric space satisfying the semiparallelogram law. While many interesting properties of the Karcher mean including Ando-Li-Mathias properties have been developed, the remarkable one is that the limit of inductive mean given by

where and denotes the residual of k mod n, coincides with the Karcher mean [14].

One can naturally ask what if we replace the original inductive mean by other geometric means. In this article we consider a new inductive mean constructed by a given mean generally on a complete metric space. We mainly show that the limit of the inductive mean constructed by a symmetric and multiplicative mean is the given mean, and see several examples for positive definite matrices.

 

2. Convergence of inductive means

Let (X, d) be a complete metric space. Let ω ∈ Δn, the simplex of positive probability vectors in ℝn convexly spanned by the unit coordinate vectors. A weighted n-mean Gn on X for n ≥ 2 is a continuous map Gn : Δn×Xn → X that is idempotent in the sense that Gn(ω; x, . . . , x) = x for all x ∈ X. A weighted n-mean Gn is symmetric or permutation invariant if Gn(ωσ;xσ) = Gn(ω; x), where ωσ = (ωσ(1), . . . ,ωσ(n)) and xσ = (xσ(1), . . . ,xσ(n)) for each permutation σ on {1, . . . , n}. A mean G on X is a sequence of means {Gn}n≥2.

For ω = (w1, . . . ,wn) ∈ Δn and x = (x1, . . . ,xn) ∈ Xn, we denote by

where the number of blocks is k. Also,

Note that ω∞ is an infinite-dimensional vector, not a probability vector.

Definition 2.1. Let ω ∈ Δn. A mean G = {Gn} on X is said to be multiplicative if for all n and x ∈ Xn,

If G is symmetric and multiplicative, then G is called intrinsic.

Let a = (a1, . . . , am) ∈ Xm and let ω = (w1, . . . ,wm) ∈ Δm. For a mean G = {Gn}n≥2, we consider the inductive mean defined as

where denotes the residual of k mod m, and

We also denote the multiple of m by m. Note that

Proposition 2.2. Let G = {Gn}n≥2 be a symmetric mean satisfying that for all n, ω = (w1, . . . ,wn) ∈ Δn and x = (x1, . . . ,xn) ∈ Xn

if x1 = · · · = xp for 1 ≤ p < n. Then G is the intrinsic mean.

Proof. Let ω = (w1, . . . ,wm) ∈ Δm and x = (x1, . . . ,xm) ∈ Xm. It is enough to show that G is multiplicative. Using the permutation invariance and the condition (3) sequentially yield that for k ≥ 2,

We now see our main result about the convergence of inductive means.

Theorem 2.3. Let G = {Gn}n≥2 be a symmetric mean satisfying the property (3). Then for any a = (a1, . . . , am) ∈ Xm and ω = (w1, . . . ,wm) ∈ Δm,

Proof. Let k ∈ ℕ. Then there are p, r ∈ ℕ ∪ {0} such that k = pm + r and 0 ≤ r < m by the division algorithm. By permutation invariance and the condition (3),

where Note that k → ∞ is equivalent to Then for s = 1, . . . , r, and for t = r + 1, . . . ,m.

Since a mean Gm is continuous, we conclude

 

3. Multivariable means of positive definite matrices

In this section we see multivariable matrix means on the open convex cone ℙ of positive definite matrices. Let A = (A1, . . . ,An) ∈ ℙn and ω = (w1, . . . ,wn) ∈ Δn.

Remark 3.1. The weighted arithmetic mean and weighted harmonic mean

are the intrinsic means. Moreover, the identity (3) is satisfied for both A and H, and hence,

Remark 3.2. The resolvent mean for parameter μ ≥ 0 is defined by

Bauschke, Moffat, and Wang [2] introduced the resolvent mean whose origin comes from the proximal average in convex analysis and optimization. Since then, many scholars have found fascinating properties such as the monotonicity for parameters and the nonexpansiveness [7, 10].

One can see the resolvent mean alternatively as

where and I is the identity matrix. So it is an intrinsic mean, and satisfies the equality (3). Therefore,

We now review a natural and attractive average, the Karcher mean, among many geometric means. Also we see some recent results and the connection with the Log-Euclidean mean.

Remark 3.3. The Karcher mean (or the least squares mean, Riemannian centroid, Cartan barycenter) is defined as the unique minimizer of the weighted sum of the squares of the Riemannian trace metric δ:

where δ(A,B) = ∥ log(A−1/2BA−1/2)∥F and ∥ · ∥F denotes the Frobenius norm. Using Karcher’s formula in [5] for the gradient of the objective function yields that the Karcher mean is the unique positive definite solution of the Karcher equation

Recently, many interesting properties of the Karcher mean have been widely studied. It has been shown in [14] that power means defined by the unique positive solution of the equation

converge to the Karcher mean as t → 0+, where A#tB = A1/2(A−1/2BA−1/2)tA1/2 is the weighted geometric mean of A and B in ℙ. Also, it has been proved in [6] that certain sequence of the Karcher means converges to the Log-Euclidean mean:

The following shows the continuity of power means P : (0, 1] × Δn × ℙn → ℙ, with respect to the Thompson metric d on ℙ given by

where ∥X∥ denotes the operator norm of X.

Lemma 3.1 ([12, Proposition 3.5]). Let ω, μ ∈ Δn and Then for s, t ∈ (0, 1]

where denotes the diameter of and

Theorem 3.2. Let and ω = (w1, . . . ,wn) ∈ Δn. Then

Proof. Let ϵ > 0 be given and let ω, μ ∈ Δn. By Lemma 3.1, there is δ > 0 such that d(ω, μ) < δ implies

For such ω, μ ∈ Δn

for t > 0 small enough, since the power means Pt converges to the Karcher mean as t → 0 in [12, 14]. By the triangle inequality, we have

Moreover, by Theorem 4.4, (P5) in [12]

So the Karcher mean Λ : Δn × ℙn → ℙ is continuous, and is a symmetric mean. Indeed, the Karcher mean satisfies all of the Ando-Li-Mathias properties [11].

Let If A1 = · · · = Ak for 1 ≤ k < n, then the Karcher equation (7) reduces to

It means that So by Theorem 2.3 it is proved. □

Proposition 3.3. Let and ω = (w1, . . . ,wn) ∈ Δn. Then the double limit converges, and

Proof. By Theorem 3.2 and the fact that the map A ∈ ℙ ↦ Am is continuous, we have

By the property (8) we get

By the definition of inductive means and the property (8), on the other hand, we obtain

where and k = pn + r for some p ∈ ℕ and 0 ≤ r < n. Note that k → ∞ is equivalent to p → ∞, and hence,

 

4. Final Remarks

Hadamard spaces are important examples of complete convex metric spaces [13, 17]. Here, a complete metric space (M, d) is called a Hadamard space if it satisfies the semiparallelogram law; for each x, y ∈ M, there exists an m ∈ M satisfying

for all z ∈ M. Such spaces are also called (global) CAT(0)-spaces or NPC (non-positively curved) spaces. The point m appearing in (9) is the unique metric midpoint between x and y. The midpoint operation gives rise to a unique minimal geodesic γx,y : [0, 1] → M for given two points x and y. We denoted by x#ty := γx,y(t) and call it the weighted geometric mean of x and y. The typical example of Hadamard space is the open convex cone ℙ of positive definite matrices with Riemannian trace metric.

On a Hadamard space (M, d), the least squares mean

exists uniquely. Motivated by Strong Law of Large Number on Hadamard spaces that established by K. Sturm [17], J. Holbrook [4], Y. Lim and M. Palfia [15] found a deterministic approximation to the least squares mean mean: For a = (a1, . . . , an) ∈ Mn,

where

and and denotes the residual of k mod n.

We naturally ask whether or not the inductive mean defined in (2) for given mean G = {Gn}n≥2 satisfies the property (11). One can see that if the mean G is symmetric and satisfies the property (3), then the property (11) does not hold. Indeed, suppose that converges to the Karcher mean as k → ∞. Then every subsequence of should converge to the Karcher mean. However, by Proposition 2.2 we have

for any p ∈ ℕ. This means that which is a contradiction to converge to the Karcher mean.