Flat connections and Wigner-Yanase-Dyson metrics

The Riemannian geometry of positive definite matrices yields state-of-the-art classification accuracy for brain-computer interface (BCI) data. The use of this framework is steadily increasing in the BCI community, sustained by its excellent classification accuracy and ability to operate transfer learning. Currently, open-source code libraries exist for the Matlab and Python programming language. Julia is a young open-source cross-platform language specifically conceived for scientific computing, which is rapidly gaining momentum in the data science community thanks to its efficiency and compatibility with the best available computing protocols. By means of this article we present and release a state-of-the-art open-source Julia package for the Riemannian geometry of positive definite matrices, named PosDefManifold. It supports nine metrics for the manifold of both real and complex positive definite matrices and includes all fundamental tools for manipulating data in them.

We develop \emph{geometric optimisation} on the manifold of Hermitian positive definite (HPD) matrices. In particular, we consider optimising two types of cost functions: (i) geodesically convex (g-convex); and (ii) log-nonexpansive (LN). G-convex functions are nonconvex in the usual euclidean sense, but convex along the manifold and thus allow global optimisation. LN functions may fail to be even g-convex, but still remain globally optimisable due to their special structure. We develop theoretical tools to recognise and generate g-convex functions as well as cone theoretic fixed-point optimisation algorithms. We illustrate our techniques by applying them to maximum-likelihood parameter estimation for elliptically contoured distributions (a rich class that substantially generalises the multivariate normal distribution). We compare our fixed-point algorithms with sophisticated manifold optimisation methods and obtain notable speedups.

Procrustes problems in Riemannian manifolds of positive definite matrices

Information geometry studies the dually flat structure of a manifold, highlighted by the generalized Pythagorean theorem. The present paper studies a class of Bregman divergences called the (ρ,τ)-divergence. A (ρ,τ) -divergence generates a dually flat structure in the manifold of positive measures, as well as in the manifold of positive-definite matrices. The class is composed of decomposable divergences, which are written as a sum of componentwise divergences. Conversely, a decomposable dually flat divergence is shown to be a (ρ,τ) -divergence. A (ρ,τ) -divergence is determined from two monotone scalar functions, ρ and τ. The class includes the KL-divergence, α-, β- and (α, β)-divergences as special cases. The transformation between an affine parameter and its dual is easily calculated in the case of a decomposable divergence. Therefore, such a divergence is useful for obtaining the center for a cluster of points, which will be applied to classification and information retrieval in vision. For the manifold of positive-definite matrices, in addition to the dually flatness and decomposability, we require the invariance under linear transformations, in particular under orthogonal transformations. This opens a way to define a new class of divergences, called the (ρ,τ) -structure in the manifold of positive-definite matrices.

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Local extrema for Procrustes problems in the set of positive definite matrices

We introduce Generalized Dictionary Learning (GDL), a simple but practical framework for learning dictionaries over the manifold of positive definite matrices. We illustrate GDL by applying it to Nearest Neighbor (NN) retrieval, a task of fundamental importance in disciplines such as machine learning and computer vision. GDL distinguishes itself from traditional dictionary learning approaches by explicitly taking into account the manifold structure of the data. In particular, GDL allows performing “sparse coding” of positive definite matrices, which enables better NN retrieval. Experiments on several covariance matrix datasets show that GDL achieves performance rivaling state-of-the-art techniques.KeywordsFace RecognitionCovariance MatriceNear NeighborOnline AlgorithmGeodesic DistanceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Riemannian metrics on positive definite matrices related to means. II

Riemannian metrics on positive definite matrices related to means

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.

We consider regularized covariance estimation in scaled Gaussian settings, e.g., elliptical distributions, compound-Gaussian processes and spherically invariant random vectors. Asymptotically in the number of samples, the classical maximum likelihood (ML) estimate is optimal under different criteria and can be efficiently computed even though the optimization is nonconvex. We propose a unified framework for regularizing this estimate in order to improve its finite sample performance. Our approach is based on the discovery of hidden convexity within the ML objective. We begin by restricting the attention to diagonal covariance matrices. Using a simple change of variables, we transform the problem into a convex optimization that can be efficiently solved. We then extend this idea to nondiagonal matrices using convexity on the manifold of positive definite matrices. We regularize the problem using appropriately convex penalties. These allow for shrinkage towards the identity matrix, shrinkage towards a diagonal matrix, shrinkage towards a given positive definite matrix, and regularization of the condition number. We demonstrate the advantages of these estimators using numerical simulations.

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.

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

A fixed point mean approximation to the Cartan barycenter of positive definite matrices

Neural tissue microstructure plays a key role in developmental, physiological and pathophysiological processes. In the continuing quest to characterize it at ever finer length scales, we use a novel diffusion tensor distribution (DTD) paradigm to probe microstructural features much smaller than the nominal MRI voxel size. We first assume the DTD is a normal tensor variate distribution constrained to lie on the manifold of positive definite matrices, characterized by a mean and covariance tensor. We then estimate the DTD using Monte Carlo signal inversion combined with parsimonious model selection framework that exploits a hierarchy of symmetries of mean and covariance tensors. High resolution multiple pulsed field gradient (mPFG) MRI measurements were performed on a homogeneous isotropic diffusion phantom (PDMS) for control, and excised visual cortex and spinal cord of macaque monkey to investigate the capabilities of DTD MRI in revealing neural tissue microstructural features using strong gradients not typically available in clinical MRI scanners. DTD-derived stains and glyphs, which disentangle size, shape, and orientation heterogeneities of microscopic diffusion tensors, are presented for all samples along with the distribution of the mean diffusivity (MD) within each voxel. We also present a new glyph to visualize the symmetric (kurtosis) and asymmetric parts of the fourth-order covariance tensor. An isotropic mean diffusion tensor and zero covariance tensor was found for the isotropic PDMS phantom, as expected, while the covariance tensor (both symmetric and asymmetric parts) for neural tissue was non-zero indicating that the kurtosis tensor may not be sufficient to fully describe the microstructure. Cortical layers were clearly delineated in the higher moments of the MD spectrum consistent with histology, and microscopic anisotropy was detected in both gray and white matter of neural tissue. DTD MRI captures crossing and splaying white matter fibers penetrating into the cortex, and skewed fiber diameter distributions in the white matter tracts within the cortex and spinal cord. DTD MRI was also shown to subsume diffusion tensor imaging (DTI) while providing additional microstructural information about tissue heterogeneity and microscopic anisotropy within each voxel.