arXiv 1507.017602015-07math.STstat.TH

对称正定矩阵空间上的黎曼高斯分布

Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

Salem Said, Lionel Bombrun, Yannick Berthoumieu, Jonathan Manton

备注：: 21 pages, 1 table; accepted for publication in IEEE Trans Inf Theory

摘要：

位于$m \times m$对称正定矩阵的空间$\mathcal{P}_{m\,}$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的应用中起着基础作用。对于这些应用，一个开放的挑战是找到一类概率分布，它能够捕获$\mathcal{P}_{m\,}$中数据的统计特性，因为它们出现在现实世界的情况下。本文通过在$\mathcal{P}_{m\,}$上引入黎曼高斯分布来应对这一挑战。Pennec在$2006$中首次考虑了这类分布。本文在现有文献中首次给出了它们的密度函数的精确表达式。这导致了两个原始的贡献。首先，详细研究了黎曼高斯分布的统计推断，揭示了极大似然估计和黎曼质心概念之间的联系，广泛应用于应用。第二，推导和实现的期望最大化算法，估计的混合黎曼高斯分布。本文采用这种新的算法，在$\mathcal{P}_{m\,}$中的数据分类，（具体地说，纹理分类的问题，在计算机视觉），表明它产生显着更好的性能，在最近的方法相比。

英文摘要：

Data which lie in the space $\mathcal{P}_{m\,}$, of $m \times m$ 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 $\mathcal{P}_{m\,}$, as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on $\mathcal{P}_{m\,}$. 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 maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and 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 $\mathcal{P}_{m\,}$, (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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