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Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

机译：对称正定矩阵空间上的黎曼高斯分布

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摘要

Data, which lie in the space Pm, 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 Pm, as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on Pm. 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 Pm, (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches.
机译：m×m对称正定矩阵的空间Pm中的数据（有时称为张量数据）在包括医学成像，计算机视觉和雷达信号处理在内的各种应用中起着基本作用。对于这些应用程序来说，一个开放的挑战是找到一类概率分布，当在实际情况下出现时，它们能够捕获Pm中数据的统计属性。本文通过介绍Pm上的黎曼高斯分布来应对这一挑战。 Pennec在2006年首次考虑了这种分布。但是，本文在现有文献中首次给出了其概率密度函数的精确表示。这导致了两个原始贡献。首先，对黎曼高斯分布的统计推断进行详细研究，揭示最大似然估计与广泛应用的黎曼质量中心概念之间的联系。其次，用于估计黎曼高斯分布的混合的期望最大化算法的推导和实现。本文将这种新算法应用于Pm中的数据分类（具体而言，应用于计算机视觉中的纹理分类问题），表明与最新方法相比，该算法产生了明显更好的性能。

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