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A Robust Distance Measure for Similarity-Based Classification on the SPD Manifold

机译：基于相似性的SPD歧管分类的鲁棒距离测量

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摘要

The symmetric positive definite (SPD) matrices, forming a Riemannian manifold, are commonly used as visual representations. The non-Euclidean geometry of the manifold often makes developing learning algorithms (e.g., classifiers) difficult and complicated. The concept of similarity-based learning has been shown to be effective to address various problems on SPD manifolds. This is mainly because the similarity-based algorithms are agnostic to the geometry and purely work based on the notion of similarities/distances. However, existing similarity-based models on SPD manifolds opt for holistic representations, ignoring characteristics of information captured by SPD matrices. To circumvent this limitation, we propose a novel SPD distance measure for the similarity-based algorithm. Specifically, we introduce the concept of point-to-set transformation, which enables us to learn multiple lower dimensional and discriminative SPD manifolds from a higher dimensional one. For lower dimensional SPD manifolds obtained by the point-to-set transformation, we propose a tailored set-to-set distance measure by making use of the family of alpha-beta divergences. We further propose to learn the point-to-set transformation and the set-to-set distance measure jointly, yielding a powerful similarity-based algorithm on SPD manifolds. Our thorough evaluations on several visual recognition tasks (e.g., action classification and face recognition) suggest that our algorithm comfortably outperforms various state-of-the-art algorithms.
机译：形成Riemannian歧管的对称正定（SPD）矩阵通常用作视觉表示。歧管的非欧几里德几何形状通常使开发学习算法（例如，分类器）困难和复杂。已显示相似性的学习的概念有效地解决了SPD歧管的各种问题。这主要是因为基于相似性的算法对几何形状不可知，并且基于相似性/距离的概念纯粹的工作。然而，在SPD歧管上的现有相似性的模型选择整体表示，忽略SPD矩阵捕获的信息的特征。为了规避本限制，我们提出了一种基于相似性的算法的新型SPD距离测量。具体而言，我们介绍了点对点变换的概念，这使我们能够从更高维度1中学习多个下维和鉴别的SPD歧管。对于通过点对点变换获得的较低尺寸SPD歧管，我们通过利用α-β发散的家族来提出定制的设定距离测量。我们进一步建议共同地学习点对点变换和设定的距离测量，在SPD歧管上产生了一种强大的相似性算法。我们对多种视觉识别任务（例如，动作分类和面部识别）的彻底评估表明，我们的算法舒适地优于各种最先进的算法。

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