A NEW PROOF TO CONSTRUCT MULTIVARIABLE GEOMETRIC MEANS BY SYMMETRIZATION†

Abstract

The original geometric mean of two positive definite operators A and B is given by A#B = A^1/2(A^-1/2BA^-1/2)^1/2A^1/2. In this article we provide a new proof to construct from the two-variable geometric mean to the multivariable mean via symmetrization introduced by Lawson and Lim [5]. Finally we provide an algorithm to find three-variable geometric mean via symmetrization, which plays an important role to construct higher-order geometric means.

1. Introduction

As a typical value of a finite number of positive real numbers, average or mean plays an important role in probability theory, statistics, and economics. For instance, the arithmetic, geometric, and harmonic means have been commonly used:

One can construct a definition of means for positive real numbers as following.

Definition 1.1. Let ℝ+ be the set of all positive real numbers. A function is called a mean of positive real numbers if

The three of the most familiar means listed above satisfy the axioms of means and hold the inequality:

This n-variable mean can be naturally defined for positive definite bounded operators. The arithmetic and harmonic means of positive definite operators can be defined as the same as those of positive real numbers, but it is not the case of the geometric mean because of non-commutativity. The purpose of this paper is to suggest a new method to construct (n + 1)-variable geometric mean from n-variable geometric mean.

Let ℙ be the open convex cone of positive definite bounded operators. For self-adjoint operators X and Y we define X ≤ Y if and only if Y − X is positive semidefinite, and X < Y if and only if Y − X is positive definite. This relation, known as the Löewner order, gives a partial order on ℙ.

 

2. Two-variable geometric mean

The original geometric mean of positive definite operators A and B

was introduced by Kubo and Ando in [4], and its several properties have been found: see the references [1] and [3]. One can naturally define the weighted geometric mean of positive definite operators A and B such as

where t ∈ [0, 1]. If A and B are not invertible, then we can take

We list some properties of the weighted geometric mean.

Lemma 2.1. Let A,B,C,D ∈ ℙ and let s, t, u ∈ [0, 1]. Then the following are satisfied.

 

3. A new proof of extension

We now present a new extension of two-variable geometric mean to multivariable geometric mean. Let Δn be a set of all positive probability vectors in ℝn, that is, ω = (w1,...,wn) ∈ Δn means that wj > 0 for all j = 1,...,n and .

Let A = (A1,...,An), B = (B1,...,Bn) ∈ ℙn and ω = (w1,...,wn) ∈ Δn. We consider an operator geometric mean G : Δn × ℙn → ℙ satisfying

(G1) G is idempotent: for any A ∈ ℙ

(G2) G is jointly homogeneous: for all aj > 0

(G3) G is permutation invariant: for any permutation σ on {1,...,n}

(G4) G is monotone: if Aj ≤ Bj for all j = 1,...,n, then

For a uniform probability vector ω = (1/n,...,1/n) we simply write G(A) = G(ω;A).

Lemma 3.1. [6, Proposition 2.5] Let G : Δn × ℙn → ℙ be jointly homogeneous and monotone. Then the following contractive property for the Thompson metric is satisfied:

where d(A,B) := ║ log(A−1/2BA−1/2)║ for the operator norm ║ㆍ║.

Let A = (A1,...,An) ∈ ℙn. For convenience, we use the notation

for any j ∈ {1,...,n}.

Theorem 3.2. Let A = (A1,A2,...,An+1) ∈ ℙn+1. Assume that the n-variable geometric mean G satisfying (G1) through (G4) exists. Consider the recursive sequences

for any j ∈ {1,...,n + 1}, where . Then sequences converge as r → ∞ and their limits equal to G(A).

For an (n + 1)-tuple A of positive definite operators, the jth sequence in the first construction is made by the operator mean G of the n-tuple obtained by removing (n + 2 − j)th component of A for any j ∈ {1,...,n + 1}. This construction gives us a new (n+1)-tuple A(1) of positive definite operators. We continue this process to get the operator means of (n + 1) variables from the same mean of n variables. The following shows how to construct three-variable geometric mean via symmetrization process.

Via Section 2 and Section 3 of [5] Lawson and Lim have introduced a way to extend higher-order means from nonexpansive and coordinatewise contractive means in a complete metric space X. See [5, Definition 3.7] for coordinatewise contractivity and [5, Definition 3.11] for nonexpansivity. Note that ℙ is the complete metric space with respect to the Thompson metric, and the mean G satisfies the nonexpansive and contractive properties by Lemma 3.1. So the map

is power convergent, which means that

for some M ∈ ℙ. Since the mean G is permutation invariant by (G3), our limit G(A) in Theorem 3.2 is the equal to M (see Remark 2.2 in [5]). While Lawson and Lim have shown it using (locally) convex hull in the geometric sense (see Proposition 3.13 in [5]), we provide a different proof of power convergence in this article.

Proof. We follow two steps.

□

Proposition 3.3. The (n + 1)-variable geometric mean G obtained by the recursive sequence (1) also satisfies the following. Let A = (A1,...,An+1) and B = (B1,...,Bn+1) ∈ ℙn+1.

Proof. These properties can be easily seen from the proof of Theorem 3.2. □

The formula of G(A1,...,An+1) is rather complicated, but it is nice with only two variables. We give an interesting property for G(A1,...,An+1) constructed by only two variables.

Corollary 3.4. Assume that A1 = ⋯ = Ak = A and Ak+1 = ⋯ = An = B for some 1 < k < n + 1. Then

where .

 

4. Numerical experiments

We give an algorithm to find three-variable geometric mean constructed by Theorem 3.2. We consider positive definite matrices instead of positive definite operators to be able to compute and show two examples using MATLAB.

Algorithm

Require: Points A0, B0 and C0 which are positive definite

Step 1: If max{║Ai − Bi║, ║Bi − Ci║, ║Ci − Ai║} ≥ ϵ, then compute

Step 2: If max{║Ai − Bi║, ║Bi − Ci║, ║Ci − Ai║} < ϵ,

In this article we are interested in the geometric mean so that we set

Example 4.1. It has been known that any 2 × 2 density matrix ρv, which is a 2-by-2 positive semidefinite Hermitian matrix with trace 1, can be parameterized by a Bloch vector v in the unit ball of ℝ3. Here,

Moreover, the 2×2 invertible density matrix described by a Bloch vector in the open unit ball B of ℝ3 plays an important role in quantum information theory. We give an example of three-variable geometric mean of 2×2 invertible density matrices.

Let ρu, ρv, and ρw be 2 × 2 density matrices parameterized by

Set ϵ = 10−4. Then by the 12th iteration we obtain

On the other hand, A. Ungar has shown in [7, Theorem 6.93] that the gyrocentroid of Bloch vectors u, v, and w in B are given by

where is the Lorentz factor. In this example we have , and so

We easily check that

Example 4.2. Let . These are positive definite matrices whose determinants are all 1. In this case we can use the following formula of two-variable geometric mean for 2 × 2 positive definite matrices A and B whose determinants are 1 (see [1, Proposition 4.1.12]):

This may reduce computing time because the geometric mean is calculated by matrix sum instead of matrix power and multiplication. Set ϵ = 10−5. Then by the 20th iteration we obtain

and we can easily verify that G(A,B,C) also has determinant 1.