Generalized BackPropagation, \'{E}tude De Cas: Orthogonality

TL;DR

This paper extends backpropagation to include layers with constrained weights using Riemannian geometry, introducing the Stiefel layer with orthogonal weights for improved deep network training and applications.

Contribution

It introduces a novel method for training deep networks with orthogonal or positive definite constraints using Riemannian optimization, including the new Stiefel layer.

Findings

Orthogonal layers improve feature learning and classification.

Stiefel layers enable efficient dimensionality reduction.

Constrained weights enhance deep network performance.

Abstract

This paper introduces an extension of the backpropagation algorithm that enables us to have layers with constrained weights in a deep network. In particular, we make use of the Riemannian geometry and optimization techniques on matrix manifolds to step outside of normal practice in training deep networks, equipping the network with structures such as orthogonality or positive definiteness. Based on our development, we make another contribution by introducing the Stiefel layer, a layer with orthogonal weights. Among various applications, Stiefel layers can be used to design orthogonal filter banks, perform dimensionality reduction and feature extraction. We demonstrate the benefits of having orthogonality in deep networks through a broad set of experiments, ranging from unsupervised feature learning to fine-grained image classification.