Expanding the Family of Grassmannian Kernels: An Embedding Perspective

TL;DR

This paper introduces new positive definite Grassmannian kernels, including universal ones, to improve the modeling of linear subspaces in visual recognition tasks, outperforming existing kernels in classification, clustering, sparse coding, and hashing.

Contribution

The paper proposes several new positive definite Grassmannian kernels, including universal kernels, enhancing the ability to model subspaces for various visual recognition applications.

Findings

New kernels outperform previous ones in classification tasks

Universal kernels provide better approximation capabilities

Enhanced performance in clustering, sparse coding, and hashing

Abstract

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g, support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show- is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing.