@gaoRobustDistanceMeasure2020

A Robust Distance Measure for Similarity-Based Classification on the SPD Manifold (2020)

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1 Abstract

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 setto-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, face recognition) suggest that our algorithm comfortably outperforms various state-of-the-art algorithms.

2 NOTES

In this paper the authors propose the of bilinear projection of SPD matrices into multiple low-dimensional manifolds, where they apply a individual learnable alpha-beta divergence to measure the distance between a pair of data. The idea of point-to-set is based on the initial multiple low-dimensional projections, such that a single matrix turns into multiple data, for two samples $X_1$ and $X_2$ there will be two sets $\{X_1^1,\dots ,X_1^n\}$ and $\{X_2^1,\dots ,X_2^n\}$ . Then, a set-to-set is done by comparing $\{X_1^1,X_2^1\},\dots ,\{X_1^n,X_2^n\}$ using the alpha-beta divergence. Then, using a third parameter matrix $M$ they combine these results into a single one, which essentially gets then all into the same manifold. Then, it is only a matter of optimizing these parameters following the objective function, hoping to reduce the distance obtained from the previous process for same class variables and increasing for different classes. Similar idea to filter-banks, but instead it relies on dimensionality reduction and distribution comparisons.