Disentangling by Subspace Diffusion

David Pfau, Irina Higgins, Alex Botev, Sebastian Racaniere

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.1 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.