Disentangling by Subspace Diffusion

TL;DR

This paper introduces GEOMANCER, a nonparametric algorithm that leverages differential geometry to unsupervisedly disentangle data manifolds by estimating invariant subspaces through diffusion, advancing understanding of geometric representation learning.

Contribution

The paper presents GEOMANCER, a novel method for symmetry-based manifold disentangling using subspace diffusion, connecting unsupervised metric learning with geometric decomposition.

Findings

GEOMANCER effectively disentangles synthetic complex manifolds.

Disentangling is feasible if the true manifold metric is known and factors have nontrivial holonomy.

The approach unifies unsupervised metric learning with geometric manifold decomposition.

Abstract

We present a novel nonparametric algorithm for symmetry-based disentangling of data manifolds, the Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER provides a partial answer to the question posed by Higgins et al. (2018): is it possible to learn how to factorize a Lie group solely from observations of the orbit of an object it acts on? We show that fully unsupervised factorization of a data manifold is possible if the true metric of the manifold is known and each factor manifold has nontrivial holonomy -- for example, rotation in 3D. Our algorithm works by estimating the subspaces that are invariant under random walk diffusion, giving an approximation to the de Rham decomposition from differential geometry. We demonstrate the efficacy of GEOMANCER on several complex synthetic manifolds. Our work reduces the question of whether unsupervised disentangling is possible to the question of whether unsupervised metric learning is possible, providing a unifying insight into the geometric nature of representation learning.