Mean Of Positive Definite Matrices Research Articles

The Ando-Li-Mathias geometric mean and infinite product of hollow matrices

In this paper we present a class of convergent infinite products of hollow matrices of the form These sequences of hollow matrices are obtained from the Ando-Li-Mathias matrix geometric mean of positive definite matrices. We further provide a three dimensional analytic manifold and an analytic self-map realizing such sequences.

Projection-free nonconvex stochastic optimization on Riemannian manifolds

Abstract We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank–Wolfe (Fw) methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: the computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.

Strong law of large numbers for the L1-Karcher mean

Sturm's strong law of large numbers in CAT(0) spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It provides an easily computable stochastic approximation based on inductive means. Convergence of a deterministic version of this approximation has been proved by Holbrook, providing his “nodice” theorem for the Karcher mean of positive definite matrices. The Karcher mean has also been extended to the infinite dimensional case of positive operators on a Hilbert space by Lawson-Lim and then to probability measures with bounded support by the second author, however the CAT(0) property of the space is lost and one defines the mean as the unique solution of a nonlinear operator equation on a convex Banach-Finsler manifold. The formulations of Sturm's strong law of large numbers and Holbrook's “nodice” approximation are natural and both conjectured to converge, however all previous techniques of their proofs break down, due to the Banach-Finsler nature of the space. In this paper we prove both conjectures by establishing the most general L1-form of Sturm's strong law of large numbers and Holbrook's “nodice” theorem in the operator norm by developing a stochastic discrete-time resolvent flow for the Karcher barycenter using its Wasserstein contraction property.

Inequalities of the Wasserstein mean with other matrix means

Recently, a new Riemannian metric and a least squares mean of positive definite matrices have been introduced. They are called the Bures–Wasserstein metric and Wasserstein mean, which are different from the Riemannian trace metric and Karcher mean. In this paper we find relationships of the Wasserstein mean with other matrix means such as the power means, harmonic mean, and Karcher mean.

On the nonlinear matrix equation [formula omitted

The nonlinear matrix equation Xp=A+MT(X#B)M, where p≥1 is a positive integer, M is an n×n nonsingular matrix, A is a positive semidefinite matrix and B is a positive definite matrix, is considered. We denote by C#D the geometric mean of positive definite matrices C and D. Based on the properties of the Thompson metric, we prove that this nonlinear matrix equation always has a unique positive definite solution and that the fixed-point iteration method can be efficiently employed to compute it. In addition, estimates of the positive definite solution and perturbation analysis are investigated. Numerical experiments are given to confirm the theoretical analysis.

Bounds for the Wasserstein mean with applications to the Lie-Trotter mean

Since barycenters in the Wasserstein space of probability distributions have been introduced, the Wasserstein metric and the Wasserstein mean of positive definite Hermitian matrices have been recently developed. In this paper, we explore some properties of Wasserstein mean of positive definite matrices and find bounds for the Wasserstein mean with respect to operator norm and Loewner order. Using bounds for the Wasserstein mean, we verify that the Wasserstein mean is the multivariate Lie-Trotter mean.

Matrix trace inequalities on Tsallis relative entropy of negative order

In this paper, we show matrix trace inequalities on the quantum Tsallis relative entropy of negative order, which includes the quasi geometric mean of positive definite matrices. Moreover, we show an estimate of the difference between two Tsallis relative entropies of negative order.

Inequalities for the Wasserstein mean of positive definite matrices

We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean.

Some matrix power and Karcher means inequalities involving positive linear maps

In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1,...,An) is an n-tuple of positive definite matrices such that 0 < m ? Ai ? M (i = 1,...,n) for some scalars m < M and ? = (w1,...,wn) is a weight vector with wi ? 0 and ?n,i=1 wi=1, then ?p (?n,i=1 wiAi)? ?p?p(Pt(?,A)) and ?p (?n,i=1 wiAi) ? ?p?p(?(?,A)), where p > 0,? = max {(M+m)2/4Mm,(M+m)2/42p Mm}, ? is a positive unital linear map and t ? [-1,1]\{0}.

Log-majorization and Lie–Trotter formula for the Cartan barycenter on probability measure spaces

We extend Ando–Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie–Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the main application of log-majorization.

A Majorization-Minimization Algorithm for Computing the Karcher Mean of Positive Definite Matrices

An algorithm for computing the Karcher mean of $n$ positive definite matrices is proposed, based on the majorization-minimization (MM) principle. The proposed MM algorithm is parameter-free, does not need to choose step sizes, and has a theoretical guarantee of asymptotic linear convergence.

The Hadamard product for the weighted Karcher means

The weighted Karcher mean of positive definite matrices is defined as the unique minimizer of the weighted sum of squares of the Riemannian distances to each of given points. Using the well-known connection between the tensor product and the Hadamard product, we show that the Hadamard product of weighted Karcher means for permuted tuples with fixed weight is bounded by the Hadamard product of given positive definite matrices. It generalizes the results for the case of two-variable geometric means and Sagae–Tanabe inductive means.

Some norm inequalities for matrix means

Inequalities for unitarily invariant norms of power means of positive definite matrices are presented.

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.

A Review of Geometric Mean of Positive Definite Matrices

A Review of Geometric Mean of Positive Definite Matrices

Operator power means due to Lawson–Lim–Pálfia for [formula omitted

For −1≤t≤1, Lim–Pálfia defined a new family of operator power means of positive definite matrices and subsequently by Lawson–Lim their notion and most of their results extend to the setting of positive invertible operators on a Hilbert space. Each of these means except t≠0 arises as a unique positive invertible solution of a non-linear operator equation and satisfies all desirable properties of power arithmetic means of positive real numbers. The purpose of this paper is to extend the range in which operator power means due to Lawson–Lim–Pálfia are defined. We investigate some properties of operator power means for t∈(−2,2)\\[−1,1].

On the Ando–Li–Mathias mean and the Karcher mean of positive definite matrices

In this paper, from the viewpoint of the Ando–Hiai inequality, we make a comparison among three geometric means: The Ando–Li–Mathias geometric mean, the Karcher mean and the chaotic geometric mean of positive definite matrices. Among others, we show complements of the -variable Ando–Hiai inequality for the Ando–Li–Mathias geometric mean by means of the Kantorovich constant.

Geometric means of structured matrices

The geometric mean of positive definite matrices is usually identified with the Karcher mean, which possesses all properties—generalized from the scalar case—a geometric mean is expected to satisfy. Unfortunately, the Karcher mean is typically not structure preserving, and destroys, e.g., Toeplitz and band structures, which emerge in many applications. For this reason, the Karcher mean is not always recommended for modeling averages of structured matrices. In this article a new definition of a geometric mean for structured matrices is introduced, its properties are outlined, algorithms for its computation, and numerical experiments are provided. In the Toeplitz case an existing mean based on the Kahler metric is analyzed for comparison.

Convex geometric means

A class of multivariable weighted geometric means of positive definite matrices admitting Jensen-type inequalities for geodesically convex functions is considered. It is shown that there are infinitely many such geometric means including the weighted inductive, Bini–Meini–Poloni and Karcher means and each of these means provides a geometric mean majorization on the space of positive definite matrices. Some connections between our geometric mean majorizations and classical results of the standard majorization of real numbers are discussed. In particular, we establish the Hardy–Littlewood–Pólya majorization theorem and also Rado’s theorem and Schur’s convexity theorem for the weighted Karcher mean.

Riemannian Distances between Geometric Means

In this paper we study the problem of finding a weighted geometric mean sufficiently close to the weighted Karcher mean of positive definite matrices. This problem arises in various numerical algorithms for computing the weighted Karcher mean and in a physical problem of averaging on the Riemannian manifold of positive definite matrices. We prove that this is possible for any “convex" geometric means satisfying the Jensen-type inequality for geodesically convex functions varying over an open set of weights in the simplex of positive probability vectors. This follows from a useful estimation of the Riemannian distance between the Karcher mean and any convex geometric means. A better upper bound between the Karcher mean and the Riemannian convex combination admitting an explicit formula is presented.