Commutativity and Disentanglement from the Manifold Perspective

TL;DR

This paper presents a manifold-based perspective on disentanglement, showing it as the discovery of local charts and linking it to commutativity of factors, with implications for learning operators and data compression.

Contribution

It introduces a manifold perspective on disentanglement, connecting it to commutativity and integrating it with group theoretic and probabilistic frameworks.

Findings

Disentanglement corresponds to local chart discovery on data manifolds.

Commutativity between factors is a key condition for disentanglement.

The framework informs approaches to learning matrix exponential operators and data compression.

Abstract

In this paper, we interpret disentanglement as the discovery of local charts of the data manifold and trace how this definition naturally leads to an equivalent condition for disentanglement: commutativity between factors of variation. We study the impact of this manifold framework to two classes of problems: learning matrix exponential operators and compressing data-generating models. In each problem, the manifold perspective yields interesting results about the feasibility and fruitful approaches their solutions. We also link our manifold framework to two other common disentanglement paradigms: group theoretic and probabilistic approaches to disentanglement. In each case, we show how these frameworks can be merged with our manifold perspective. Importantly, we recover commutativity as a central property in both alternative frameworks, further highlighting its importance in disentanglement.