A Riemannian Network for SPD Matrix Learning

TL;DR

This paper introduces a novel Riemannian deep network architecture for learning SPD matrices, incorporating bilinear, eigenvalue rectification, and logarithm layers, trained with specialized backpropagation on Stiefel manifolds, outperforming existing methods.

Contribution

It presents the first Riemannian network architecture for SPD matrix learning with new layers and a specialized training method, advancing deep learning on Riemannian manifolds.

Findings

Outperforms existing SPD matrix learning methods in visual classification tasks

Eases training of deep models on SPD matrices with a new backpropagation approach

Demonstrates effective non-linear SPD matrix transformations within deep networks

Abstract

Symmetric Positive Definite (SPD) matrix learning methods have become popular in many image and video processing tasks, thanks to their ability to learn appropriate statistical representations while respecting Riemannian geometry of underlying SPD manifolds. In this paper we build a Riemannian network architecture to open up a new direction of SPD matrix non-linear learning in a deep model. In particular, we devise bilinear mapping layers to transform input SPD matrices to more desirable SPD matrices, exploit eigenvalue rectification layers to apply a non-linear activation function to the new SPD matrices, and design an eigenvalue logarithm layer to perform Riemannian computing on the resulting SPD matrices for regular output layers. For training the proposed deep network, we exploit a new backpropagation with a variant of stochastic gradient descent on Stiefel manifolds to update the structured connection weights and the involved SPD matrix data. We show through experiments that the proposed SPD matrix network can be simply trained and outperform existing SPD matrix learning and state-of-the-art methods in three typical visual classification tasks.