This work proposes a practical method for disentangling, i.e., Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER does not learn a global nonlinear embedding, instead, it learns a set of subspaces to assign to each point, where each subspace is the tangent space of one disentangled submanifold. Thus, GEOMANCER can be used to disentangle manifolds for which there may not be a global axis-aligned coordinate system. Experimental results on both synthetic data and Stanford 3D data are included in this paper.

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 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.

This work proposes a practical method for disentangling, i.e., Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER does not learn a global nonlinear embedding, instead, it learns a set of subspaces to assign to each point, where each subspace is the tangent space of one disentangled submanifold. Thus, GEOMANCER can be used to disentangle manifolds for which there may not be a global axis-aligned coordinate system. Experimental results on both synthetic data and Stanford 3D data are included in this paper.