Quantifying and Learning Linear Symmetry-Based Disentanglement

TL;DR

This paper introduces a new metric for quantifying Linear Symmetry-Based Disentanglement (LSBD) and proposes LSBD-VAE, a semi-supervised method to learn LSBD representations, demonstrating its effectiveness over existing methods.

Contribution

The paper formalizes LSBD, proposes a metric $\\mathcal{D}_\mathrm{LSBD}$ for quantification, and develops LSBD-VAE, a semi-supervised learning approach for LSBD representations.

Findings

Common VAE methods do not learn LSBD representations.

LSBD-VAE and recent methods can learn LSBD with limited supervision.

LSBD representations satisfy properties of existing disentanglement metrics.

Abstract

The definition of Linear Symmetry-Based Disentanglement (LSBD) formalizes the notion of linearly disentangled representations, but there is currently no metric to quantify LSBD. Such a metric is crucial to evaluate LSBD methods and to compare to previous understandings of disentanglement. We propose $\mathcal{D}_\mathrm{LSBD}$, a mathematically sound metric to quantify LSBD, and provide a practical implementation for $\mathrm{SO}(2)$ groups. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised method to learn LSBD representations. We demonstrate the utility of our metric by showing that (1) common VAE-based disentanglement methods don't learn LSBD representations, (2) LSBD-VAE as well as other recent methods can learn LSBD representations, needing only limited supervision on transformations, and (3) various desirable properties expressed by existing disentanglement metrics are also achieved by LSBD representations.