Random projections on manifolds of Symmetric Positive Definite matrices for image classification

Abstract

Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent spaces, or (2) embedding into Reproducing Kernel Hilbert Spaces (RKHS). While embedding into tangent spaces allows the use of existing Euclidean-based learning algorithms, manifold shape is only approximated which can cause loss of discriminatory information. The RKHS approach retains more of the manifold structure, but may require non-trivial effort to kernelise Euclidean-based learning algorithms. In contrast to the above approaches, in this paper we offer a novel solution that allows SPD matrices to be used with unmodified Euclidean-based learning algorithms, with the true manifold shape well-preserved. Specifically, we propose to project SPD matrices using a set of random projection hyperplanes over RKHS into a random projection space, which leads to representing each matrix as a vector of projection coefficients. Experiments on face recognition, person re-identification and texture classification show that the proposed approach outperforms several recent methods, such as Tensor Sparse Coding, Histogram Plus Epitome, Riemannian Locality Preserving Projection and Relational Divergence Classification.

Fast and accurate image and video analysis on Riemannian manifolds

“…(2.8)) is used for this dataset. as this has been shown to be effective in various classification problem domains [5,6].…”

“…Log-Euclidean space [139]; ROSE: random projection space with standard Gaussian hyperplanes [5]; RKHS: Reproducing Kernel Hilbert Space generated by manifold- …”

“…We first adapt the random projection method proposed in our previous work [5] to this framework and rename it as Kernelised Gaussian Random Projection (KGRP). Also, we further develop two projection hyperplane generation algorithms which also follow in our framework: Kernelised…”

“…It is often observed that the right choice of kernel could significantly improve the performance [76]. Furthermore, in general, kernel inner products with specified metrics have much less computational complexity than geodesic distances [5,127]. With these properties, kernel-based approaches could be suitable to address issues suffered in both the in-trinsic approach and the Log Euclidean approach.…”

“…We note that the above formulation was first described for developing the kernelise locality sensitive hashing method in Euclidean scenarios [83]. We then adapted the method in our previous work [5] to perform random projection on SPD manifolds for classification purposes. In this chapter, we further apply the method for clustering on Riemannian manifold problems.…”