Collaborative Representation for SPD Matrices with Application to Image-Set Classification

TL;DR

This paper introduces a novel collaborative representation method for Symmetric Positive Definite (SPD) matrices, enabling effective image-set classification by embedding SPD data into tangent space and RKHS, with demonstrated superior performance on benchmarks.

Contribution

It extends collaborative representation to SPD manifolds using tangent space and Riemannian kernel embeddings, addressing nonlinear variational information in image-set classification.

Findings

Effective classification on four benchmark datasets.

Improved handling of nonlinear variational information.

Successful embedding of SPD matrices into tangent space and RKHS.

Abstract

Collaborative representation-based classification (CRC) has demonstrated remarkable progress in the past few years because of its closed-form analytical solutions. However, the existing CRC methods are incapable of processing the nonlinear variational information directly. Recent advances illustrate that how to effectively model these nonlinear variational information and learn invariant representations is an open challenge in the community of computer vision and pattern recognition To this end, we try to design a new algorithm to handle this problem. Firstly, the second-order statistic, i.e., covariance matrix is applied to model the original image sets. Due to the space formed by a set of nonsingular covariance matrices is a well-known Symmetric Positive Definite (SPD) manifold, generalising the Euclidean collaborative representation to the SPD manifold is not an easy task. Then, we devise two strategies to cope with this issue. One attempts to embed the SPD manifold-valued data representations into an associated tangent space via the matrix logarithm map. Another is to embed them into a Reproducing Kernel Hilbert Space (RKHS) by utilizing the Riemannian kernel function. After these two treatments, CRC is applicable to the SPD manifold-valued features. The evaluations on four banchmarking datasets justify its effectiveness.