Matrix Geometric Mean Research Articles

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Geometric means of quasi-Toeplitz matrices

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $$A=(a_{i,j})_{i,j=1,2,\ldots }$$ of the form $$A=T(a)+E$$ , where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series $$\sum _{k=-\infty }^{\infty } a_k e^{{{\mathfrak {i}}}k t}$$ , in the sense that $$(T(a))_{i,j}=a_{j-i}$$ . If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on $$\ell ^2$$ . We prove that if $$a_1,\ldots ,a_p$$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then matrix geometric means, such as the ALM, the NBMP and the weighted mean of quasi-Toeplitz positive definite matrices associated with $$a_1,\ldots ,a_p$$ , are quasi-Toeplitz matrices associated with the function $$(a_1\cdots a_p)^{\frac{1}{p}}$$ , which differ only by the compact correction. We introduce numerical algorithms for their computation and show by numerical tests that these operator means can be effectively approximated numerically.

Finding the geometric mean for two Hermitian matrices by an efficient iteration method

In this work, an iterative method is developed to find the matrix sign function. It is discussed in detail that the method is novel in comparison to its competitors from the Padé family of methods for the same purpose. The convergence analysis and the stability of the method are given. The method is then extended to compute the matrix geometric mean of two Hermitian matrices having positive definiteness. This could be employed for solving a Riccati matrix equation, which is of practical importance. Some numerical results are furnished at the end to reveal the effectiveness of the proposed iteration method.

Riemannian Optimization via Frank-Wolfe Methods

We study projection-free methods for constrained Riemannian optimization. In particular, we propose a Riemannian Frank-Wolfe (RFW) method that handles constraints directly, in contrast to prior methods that rely on (potentially costly) projections. We analyze non-asymptotic convergence rates of RFW to an optimum for geodesically convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize RFW to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian “linear” oracle required by RFW admits a closed form solution; this result may be of independent interest. We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods, and observe that RFW performs competitively on the task of computing Riemannian centroids.

The quantum ♮α-Rényi divergence of real order

There are many proposals for defining a quantum version of the Rényi divergence on the differentiable manifold P of positive definite matrices. The geodesic connecting two points on P is a weighted matrix geometric mean, and its tangent vector is the relative operator entropy. It is a natural idea to use the relative operator entropy to describe the difference between two points on P. The quantum version of the classical results of Rényi divergence does not always hold due to its non-commutativity. Due to this difficulty, we use the upper bound and the lower one of the spectrum of positive definite matrices to estimate the order relation between quantum divergences, which is called the Mond Pečarić method in operator theory. In this paper, we show fundamental properties of the ♮α-Rényi divergence of real order α∈R, including monotonicity of order, the relation of max- and min-divergence, data processing inequality, bound estimates, and convexity. We also estimate the difference without the α–z-Rényi divergence and ♮α-Rényi divergence in terms of the generalized Kantorovich constant and the Specht ratio.

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.

Low Phase-Rank Approximation

In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. It is well known that a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix being defined as the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with the matrix arithmetic mean, it turns out that a natural parallel for the low phase-rank approximation problem is to use the matrix geometric mean to measure the approximation error. Importantly, we derive a majorization inequality between the phases of the geometric mean and the arithmetic mean of the phases, similarly to the Ky-Fan inequality for eigenvalues of Hermitian matrices. A characterization of the solutions to the proposed problem, with the same flavor as the Schmidt-Mirsky theorem, is then obtained in the case where both the objective matrix and the approximant are restricted to be positive-imaginary. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary.

Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means

We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function (1−det⁡(X)β)/β. We give explicit expressions for the geodesics and distance function, under suitable conditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As β tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric associated with the matrix geometric mean; we show that the geodesic and the distance associated with the power potential converge to the weighted matrix geometric mean and the distance associated with the logarithmic potential, respectively.

Computing the matrix geometric mean: Riemannian versus Euclidean conditioning, implementation techniques, and a Riemannian BFGS method

SummaryThis paper addresses the problem of computing the Riemannian center of mass of a collection of symmetric positive definite matrices. We show in detail that the condition number of the Riemannian Hessian of the underlying optimization problem is never very ill conditioned in practice, which explains why the Riemannian steepest descent approach has been observed to perform well. We also show theoretically and empirically that this property is not shared by the Euclidean Hessian. We then present a limited‐memory Riemannian BFGS method to handle this computational task. We also provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. Through empirical results and a computational complexity analysis, we demonstrate the robust behavior of the limited‐memory Riemannian BFGS method and the efficiency of our implementation when compared to state‐of‐the‐art algorithms.

Matrix Hölder-McCarthy inequality via matrix geometric means

In this paper, by virtue of an expression of matrix geometric means for positive semidefinite matrices via the Moore-Penrose inverse, we show matrix versions of the Holder-McCarthy inequality, the Holder inequality and quasi-arithmetic power means via matrix geometric means, and their reverses for positive definite matrices via the generalized Kantorovich constant.

A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.

Matrix Wielandt inequality via the matrix geometric mean

In this paper, by virtue of the matrix geometric mean and the polar decomposition, we present new Wielandt type inequalities for matrices of any size. To this end, based on results due to J.I. Fujii, we reform a matrix Cauchy–Schwarz inequality, which differs from ones due to Marshall and Olkin. As an application, we show a new block matrix version of Wielandt type inequalities under the block rank additivity condition.

Matrix geometric means based on shuffled inductive sequences

We propose a new deterministic sequence converging to the least squares mean of N symmetric positive definite (SPD) matrices. By “least squares mean”, we refer to the Riemannian barycenter with respect to the natural metric (also known as the trace metric or affine-invariant metric) on the set Pn of n×n SPD matrices. In some papers, this mean is also referred to as the Karcher mean. The sequence we propose belongs to the family of inductive sequences, obtained by letting a point take successive steps towards the data points, with step lengths progressively diminishing to zero. We use the word “inductive” since each point of those sequences can be seen as the inductive mean of a well-chosen set of points containing multiple replications of the data points. We show that visiting the data points in a cyclic order usually results in a slow convergence, and remedy this weakness by reordering repeatedly the data points using a shuffling algorithm. We prove the convergence of the resulting inductive sequence to the least squares mean of the data in the general framework of NPC spaces (complete metric spaces with non-positive curvature), which includes the set Pn. We also illustrate numerically on Pn that some points of this sequence are fairly accurate estimates of the least squares mean, while being considerably cheaper to compute, and perform well as initializers for state-of-the-art algorithms. To reduce further the computation time, we finally exploit inductive sequences to produce a family of parallelizable algorithms for estimating the least squares mean.

The Riemannian Barzilai–Borwein method with nonmonotone line search and the matrix geometric mean computation

The Riemannian Barzilai–Borwein method with nonmonotone line search and the matrix geometric mean computation

Lieb's concavity theorem, matrix geometric means, and semidefinite optimization

A famous result of Lieb establishes that the map (A,B)↦tr[K⁎A1−tKBt] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t∈[0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t∈[0,1]. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.

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.

Weighted multivariable operator means of positive definite operators

In this paper we present a new weighted, multivariable operator mean of positive definite operators over an arbitrary Hilbert space which provides us the first generally applicable extension of the classical Kubo–Ando theory of 2-variable operator means. The construction is a weighted extension of the Bini–Meini–Poloni symmetrization process originally given for the matrix geometric mean. Here to be able to consider such an iterative procedure, we need a weighted version of every Kubo–Ando mean in two variables. Therefore we also give a new construction for two arbitrary positive operators on a possibly infinite dimensional Hilbert space that provides weighted counterparts to every (not-necessarily symmetric) Kubo–Ando mean and also agrees with the most well known weighted operator means.

NO DICE THEOREM ON SYMMETRIC CONES

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number \cite{LL1, BK}. A natural question concerned with the S.L.L.N is so called {\it the no dice conjecture}. It is a problem to make a construction of deterministic sequences converging to the least squares mean without any probabilistic arguments. Very recently, Holbrook \cite{Hol} gave an affirmative answer to the conjecture in the space of positive definite matrices. In this paper, inspired by the work of Holbrook \cite{Hol} and the fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we establish the no dice theorem on general symmetric cones.

On the semidefinite representation of real functions applied to symmetric matrices

We present a new semidefinite representation for the trace of a real function f applied to symmetric matrices, when a semidefinite representation of the convex (or concave) function f is known. Our construction is intuitive, and yields a representation that is more compact than the previously known one. We also show with the help of matrix geometric means and a Riemannian metric over the set of positive definite matrices that for a rational exponent p in the interval (0,1], the matrix X raised to p is the largest element of a set represented by linear matrix inequalities. This result further generalizes to the case of the matrix A♯pB, which is the point of coordinate p on the geodesic from A to B. We give numerical results for a problem inspired from the theory of experimental designs, which show that the new semidefinite programming formulation can yield an important speed-up factor.

Operator entropy and fidelity associated with the geometric mean

We consider operator versions of quantum Tsallis relative entropy that involve the weighted matrix geometric mean and establish relationships with the fidelity of a pure state and an arbitrary state. We define the symmetrized operator entropy based on the Tsallis relative operator entropy and study its properties.