Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

• Published in: IEEE transactions on information theory Vol. 63; no. 4; pp. 2153 - 2170
• Main Authors: Said, Salem, Bombrun, Lionel, Berthoumieu, Yannick, Manton, Jonathan H.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-04-2017; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Algorithms; Classification; Computer Science; Computer vision; Economic models; Estimating techniques; expectation-maximisation; Gaussian distribution; Mathematical analysis; Matrix methods; Maximum likelihood estimation; Measurement; Medical imaging; Probability density function; Probability density functions; Probability distribution; Radar imaging; Riemannian metric; Signal and Image Processing; Signal processing; Statistical analysis; Statistical inference; Symmetric matrices; Symmetric positive definite matrices; tensor; Tensors; texture
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2017.2653803

Data, which lie in the space P m , of m × m symmetric positive definite matrices, (sometimes called tensor data), play a fundamental role in applications, including medical imaging, computer vision, and radar signal processing. An open challenge, for these applications, is to find a class of probability distributions, which is able to capture the statistical properties of data in P m , as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on P m . Distributions of this kind were first considered by Pennec in 2006. However, the present paper gives an exact expression of their probability density function for the first time in existing literature. This leads to two original contributions. First, a detailed study of statistical inference for Riemannian Gaussian distributions, uncovering the connection between the maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and the implementation of an expectation-maximisation algorithm, for the estimation of mixtures of Riemannian Gaussian distributions. The paper applies this new algorithm, to the classification of data in P m , (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches.