Manifold Of Symmetric Positive Definite Matrices Research Articles

Ricci flow-based brain surface covariance descriptors for diagnosing Alzheimer’s disease

Automated feature extraction from MRI brain scans and diagnosis of Alzheimer’s disease are ongoing challenges. With advances in 3D imaging technology, 3D data acquisition is becoming more viable and efficient than its 2D counterpart. Rather than using feature-based vectors, in this paper, for the first time, we suggest a pipeline to extract novel covariance-based descriptors from the cortical surface using the Ricci energy optimization. The covariance descriptors are components of the nonlinear manifold of symmetric positive-definite matrices, thus we focus on using the Gaussian radial basis function to apply manifold-based classification to the 3D shape problem. Applying this novel signature to the analysis of abnormal cortical brain morphometry allows for diagnosing Alzheimer’s disease. Experimental studies performed on about two hundred 3D MRI brain models, gathered from Alzheimer’s Disease Neuroimaging Initiative (ADNI) dataset demonstrate the effectiveness of our descriptors in achieving remarkable classification accuracy.

Alpha Procrustes metrics between positive definite operators: A unifying formulation for the Bures-Wasserstein and Log-Euclidean/Log-Hilbert-Schmidt metrics

This work presents a parametrized family of distances, namely the Alpha Procrustes distances, on the set of symmetric, positive definite (SPD) matrices. The Alpha Procrustes distances provide a unified formulation encompassing both the Bures-Wasserstein and Log-Euclidean distances between SPD matrices. We show that the Alpha Procrustes distances are the Riemannian distances corresponding to a family of Riemannian metrics on the manifold of SPD matrices, which encompass both the Log-Euclidean and Wasserstein Riemannian metrics. This formulation is then generalized to the set of positive definite Hilbert-Schmidt operators on a Hilbert space, unifying the infinite-dimensional Bures-Wasserstein and Log-Hilbert-Schmidt distances. In the setting of reproducing kernel Hilbert spaces (RKHS) covariance operators, we obtain closed form formulas for all the distances via the corresponding kernel Gram matrices. From a statistical viewpoint, the Alpha Procrustes distances give rise to a parametrized family of distances between Gaussian measures on Euclidean space, in the finite-dimensional case, and separable Hilbert spaces, in the infinite-dimensional case, encompassing the 2-Wasserstein distance, with closed form formulas via Gram matrices in the RKHS setting. The presented formulations are new both in the finite and infinite-dimensional settings.

Geometry-aware manipulability learning, tracking, and transfer

Body posture influences human and robot performance in manipulation tasks, as appropriate poses facilitate motion or the exertion of force along different axes. In robotics, manipulability ellipsoids arise as a powerful descriptor to analyze, control, and design the robot dexterity as a function of the articulatory joint configuration. This descriptor can be designed according to different task requirements, such as tracking a desired position or applying a specific force. In this context, this article presents a novel manipulability transfer framework, a method that allows robots to learn and reproduce manipulability ellipsoids from expert demonstrations. The proposed learning scheme is built on a tensor-based formulation of a Gaussian mixture model that takes into account that manipulability ellipsoids lie on the manifold of symmetric positive-definite matrices. Learning is coupled with a geometry-aware tracking controller allowing robots to follow a desired profile of manipulability ellipsoids. Extensive evaluations in simulation with redundant manipulators, a robotic hand and humanoids agents, as well as an experiment with two real dual-arm systems validate the feasibility of the approach.

Riemannian Optimal Identification Method for Linear Systems With Symmetric Positive-Definite Matrix

This article develops identification methods for linear continuous-time symmetric systems, such as electrical network systems, multiagent network systems, and temperature dynamics in buildings. To this end, we formulate three system identification problems for the corresponding discrete-time systems. The first is a least-squares problem in which we wish to minimize the sum of squared errors between the true and model outputs on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. In the second problem, to reduce the search dimensions, the product manifold is replaced with the quotient set under a specified group action by the orthogonal group. In the third problem, the manifold of symmetric positive-definite matrices in the first problem is replaced by the manifold of matrices with only positive diagonal elements. In particular, we examine the quotient geometry in the second problem. We propose Riemannian conjugate gradient methods for the three problems, and select initial points using a popular subspace method. The effectiveness of our proposed methods is demonstrated through numerical simulations and comparisons with the Gauss–Newton method, which is one of the most popular approach for solving least-squares problems.

Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation

In this paper, we consider the problem of domain adaptation. We propose to view the data through the lens of covariance matrices and present a method for domain adaptation using parallel transport on the cone manifold of symmetric positive-definite matrices. We provide rigorous analysis using Riemanninan geometry, illuminating the theoretical guarantees and benefits of the presented method. In addition, we demonstrate these benefits using experimental results on simulations and real-measured data.

Structure-Preserving $H^2$ Optimal Model Reduction Based on the Riemannian Trust-Region Method

This paper studies stability- and symmetry-preserving H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> optimal model reduction problems of linear systems, which include linear gradient systems as a special case. The problem is formulated as a nonlinear optimization problem on the product manifold of the manifold of symmetric positive-definite matrices and two Euclidean spaces. To solve the problem by using the trust-region method, the gradient and Hessian of the objective function are derived. Furthermore, it is shown that if we restrict our systems to gradient systems, the gradient and Hessian can be obtained more efficiently. More concretely, by symmetry, we can reduce linear matrix equations to be solved. In addition, by a simple example, we show that the solutions to our problem and a similar problem in some literature works are not unique, and the solution sets of both problems do not contain each other in general. Also, it is revealed that the attained optimal values do not coincide. Numerical experiments show that the proposed method gives a reduced system with the same structure with the original system although the balanced truncation method does not.

A Fanning Scheme for the Parallel Transport along Geodesics on Riemannian Manifolds

Parallel transport on Riemannian manifolds allows one to connect tangent spaces at different points in an isometric way and is therefore of importance in many contexts, such as for statistics on manifolds. The existing methods to compute parallel transport require either the computation of Riemannian logarithms, such as the Schild's ladder, or the Christoffel symbols. The Logarithm is rarely given in closed form, and therefore costly to compute whereas the number of Christoffel symbols explodes with the dimension of the manifold, making both these methods intractable. From an identity between parallel transport and Jacobi fields, we propose a numerical scheme to approximate the parallel transport along a geodesic. We find and prove an optimal convergence rate for the scheme, which is equivalent to Schild's ladder's. We investigate potential variations of the scheme and give experimental results on the Euclidean two-sphere and on the manifold of symmetric positive-definite matrices.

A closed-form unsupervised geometry-aware dimensionality reduction method in the Riemannian Manifold of SPD matrices.

Riemannian geometry has been found accurate and robust for classifying multidimensional data, for instance, in brain-computer interfaces based on electroencephalography. Given a number of data points on the manifold of symmetric positive-definite matrices, it is often of interest to embed these points in a manifold of smaller dimension. This is necessary for large dimensions in order to preserve accuracy and useful in general to speed up computations. Geometry-aware methods try to accomplish this task while respecting as much as possible the geometry of the original data points. We provide a closed-form solution for this problem in a fully unsupervised setting. Through the analysis of three brain-computer interface data bases we show that our method allows substantial dimensionality reduction without affecting the classification accuracy.

Nonlinear damped oscillators on Riemannian manifolds: Fundamentals

The classical theory of mass-spring-damper-type dynamical systems on the ordinary flat space ℝ3 may be generalized to higher-dimensional Riemannian manifolds by reformulating the basic underlying physical principles through differential geometry. Nonlinear dynamical systems have been studied in the scientific literature because they arise naturally from the modeling of complex physical structures and because such dynamical systems constitute the basis for several modern applications such as the secure transmission of information. The flows of nonlinear dynamical systems may evolve over time in complex, non-repeating (although deterministic) patterns. The focus of the present paper is on formulating the general equations that describe the dynamics of a point-wise particle sliding on a Riemannian manifold in a coordinate-free manner. The paper shows how the equations particularize in the case of some manifolds of interest in the scientific literature, such as the Stiefel manifold and the manifold of symmetric positive-definite matrices.

A Tractable State-Space Model for Symmetric Positive-Definite Matrices

The Bayesian analysis of a state-space model includes computing the posterior distribution of the system’s parameters as well as its latent states. When the latent states wander around Rn there are several well-known modeling components and computational tools that may be profitably combined to achieve this task. When the latent states are constrained to a strict subset of Rn these models and tools are either impaired or break down completely. State-space models whose latent states are covariance matrices arise in finance and exemplify the challenge of devising tractable models in the constrained setting. To that end, we present a state-space model whose observations and latent states take values on the manifold of symmetric positive-definite matrices and for which one may easily compute the posterior distribution of the latent states and the system’s parameters as well as filtered distributions and one-step ahead predictions. Employing the model within the context of finance, we show how one can use realized covariance matrices as data to predict latent time-varying covariance matrices. This approach out-performs factor stochastic volatility.

Learning general Gaussian kernel hyperparameters of SVMs using optimization on symmetric positive-definite matrices manifold

We propose a new method for general Gaussian kernel hyperparameter optimization for support vector machines classification. The hyperparameters are constrained to lie on a differentiable manifold. The proposed optimization technique is based on a gradient-like descent algorithm adapted to the geometrical structure of the manifold of symmetric positive-definite matrices. We compare the performance of our approach with the classical support vector machine for classification and with other methods of the state of the art on toy data and on real world data sets.

Visualization of Riemannian-manifold-valued elements by multidimensional scaling

The present contribution suggests to utilize a multidimensional scaling algorithm as a visualization tool for high-dimensional smoothly constrained learnable-system's patterns that lie on Riemannian manifolds. Such visualization tool proves useful in machine learning whenever learning/adaptation algorithms insist on high-dimensional Riemannian parameter manifolds. In particular, the manuscript describes the cases of interest in the recent scientific literature that the parameter space is the set of special orthogonal matrices, the unit hypersphere and the manifold of symmetric positive-definite matrices. The paper also recalls the notion of multidimensional scaling and discusses its algorithmic implementation. Some numerical experiments performed on toy problems help the readers to get acquainted with the problem at hand, while experiments performed on independent component analysis data as well as averaging data show the usefulness of the proposed visualization tool.