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

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 wellknown 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 manifoldvalued 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.

Multi-Manifold Attention for Vision Transformers

“…Yu et al proposed the contour covariance that lies on the SPD manifold as a region descriptor for accurate image classification [32]. Similarly, Chu et al proposed the modelling of image sets with covariance matrices for improved classification performance [33]. The importance and usefulness of feature modelling on the SPD manifold can be further highlighted from the design of novel deep networks and network layers, such as Variational Autoencoders [34], LSTMs [35], GRUs [36] and mapping and pooling layers [37] to handle and learn from features on the SPD manifold.…”