Disentangling images with Lie group transformations and sparse coding

TL;DR

This paper introduces a Bayesian generative model that unsupervisedly learns to disentangle spatial patterns and their continuous transformations in images, leveraging Lie groups and sparse coding to capture symmetries and transformations.

Contribution

It combines Lie group theory with sparse coding in a Bayesian framework to learn shape components and transformations without supervision, capturing symmetries in natural signals.

Findings

Successfully recovers transformations in controlled MNIST experiments.

Learns digit shapes and natural transformations like shearing and stretching.

Demonstrates unsupervised disentanglement of spatial patterns and transformations.

Abstract

Discrete spatial patterns and their continuous transformations are two important regularities contained in natural signals. Lie groups and representation theory are mathematical tools that have been used in previous works to model continuous image transformations. On the other hand, sparse coding is an important tool for learning dictionaries of patterns in natural signals. In this paper, we combine these ideas in a Bayesian generative model that learns to disentangle spatial patterns and their continuous transformations in a completely unsupervised manner. Images are modeled as a sparse superposition of shape components followed by a transformation that is parameterized by n continuous variables. The shape components and transformations are not predefined, but are instead adapted to learn the symmetries in the data, with the constraint that the transformations form a representation of an n-dimensional torus. Training the model on a dataset consisting of controlled geometric transformations of specific MNIST digits shows that it can recover these transformations along with the digits. Training on the full MNIST dataset shows that it can learn both the basic digit shapes and the natural transformations such as shearing and stretching that are contained in this data.