Topographic VAEs learn Equivariant Capsules

TL;DR

This paper introduces the Topographic VAE, a deep generative model with topographically organized latent variables that learns to encode salient features and approximate equivariance from sequences, improving transformation handling.

Contribution

The paper presents the Topographic VAE, which combines topographic organization with equivariance learning, enabling unsupervised learning of equivariant features directly from data sequences.

Findings

Organizes latent space by salient features like digit class, width, style.

Learns approximately equivariant features ('capsules') from sequences.

Achieves higher likelihood on transformed test sequences.

Abstract

In this work we seek to bridge the concepts of topographic organization and equivariance in neural networks. To accomplish this, we introduce the Topographic VAE: a novel method for efficiently training deep generative models with topographically organized latent variables. We show that such a model indeed learns to organize its activations according to salient characteristics such as digit class, width, and style on MNIST. Furthermore, through topographic organization over time (i.e. temporal coherence), we demonstrate how predefined latent space transformation operators can be encouraged for observed transformed input sequences -- a primitive form of unsupervised learned equivariance. We demonstrate that this model successfully learns sets of approximately equivariant features (i.e. "capsules") directly from sequences and achieves higher likelihood on correspondingly transforming test sequences. Equivariance is verified quantitatively by measuring the approximate commutativity of the inference network and the sequence transformations. Finally, we demonstrate approximate equivariance to complex transformations, expanding upon the capabilities of existing group equivariant neural networks.