Building Deep Networks on Grassmann Manifolds

TL;DR

This paper introduces a novel deep learning architecture tailored for data on Grassmann manifolds, enabling effective learning on structured geometric data in visual recognition tasks.

Contribution

It generalizes Euclidean deep networks to Grassmann manifolds by designing specialized layers and a manifold-aware training algorithm, advancing geometric deep learning.

Findings

Outperforms existing Grassmann learning methods

Achieves results comparable to state-of-the-art approaches

Demonstrates effectiveness on multiple visual recognition tasks

Abstract

Learning representations on Grassmann manifolds is popular in quite a few visual recognition tasks. In order to enable deep learning on Grassmann manifolds, this paper proposes a deep network architecture by generalizing the Euclidean network paradigm to Grassmann manifolds. In particular, we design full rank mapping layers to transform input Grassmannian data to more desirable ones, exploit re-orthonormalization layers to normalize the resulting matrices, study projection pooling layers to reduce the model complexity in the Grassmannian context, and devise projection mapping layers to respect Grassmannian geometry and meanwhile achieve Euclidean forms for regular output layers. To train the Grassmann networks, we exploit a stochastic gradient descent setting on manifolds of the connection weights, and study a matrix generalization of backpropagation to update the structured data. The evaluations on three visual recognition tasks show that our Grassmann networks have clear advantages over existing Grassmann learning methods, and achieve results comparable with state-of-the-art approaches.