Topographic VAEs learn Equivariant Capsules (Machine Learning Research Paper Explained)

#tvae #topographic #equivariant

Variational Autoencoders model the latent space as a set of independent Gaussian random variables, which the decoder maps to a data distribution. However, this independence is not always desired, for example when dealing with video sequences, we know that successive frames are heavily correlated. Thus, any latent space dealing with such data should reflect this in its structure. Topographic VAEs are a framework for defining correlation structures among the latent variables and induce equivariance within the resulting model. This paper shows how such correlation structures can be built by correctly arranging higher-level variables, which are themselves independent Gaussians.

OUTLINE:
0:00 - Intro
1:40 - Architecture Overview
6:30 - Comparison to regular VAEs
8:35 - Generative Mechanism Formulation
11:45 - Non-Gaussian Latent Space
17:30 - Topographic Product of Student-t
21:15 - Introducing Temporal Coherence
24:50 - Topographic VAE
27:50 - Experimental Results
31:15 - Conclusion & Comments

Paper: https://arxiv.org/abs/2109.01394
Code: https://github.com/akandykeller/topog...

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.

Authors: T. Anderson Keller, Max Welling

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