Quantifying and Learning Linear Symmetry-Based Disentanglement

Loek Tonnaer, Luis Armando Pérez Rey, Vlado Menkovski, Mike Holenderski, Jacobus Willem Portegies

Keywords: Representation Learning, Disentanglement

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. 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 \emph{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.

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TL;DR: We propose a metric to quantify linearly symmetry-based disentangled representations, as well as a method to learn such representations with limited supervision.