Chapter 4 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) The definition of Linear Symmetry-Based Disentanglement (LSBD) formalises the notion that representations should reflect underlying symmetries of data. However, this definition does not include a metric to quantify how well representations are disentangled. Such a metric is crucial to evaluate LSBD methods and to compare to previous understandings of disentanglement. In this chapter, we present DLSBD, a mathematically sound metric to quantify Linear Symmetry-Based Disentanglement (LSBD); and give a practical implementation for SO(2), a common group structure that models 2D rotations. Furthermore, from this metric we derive LSBD-VAE, a semi-supervised method to learn LSBD representations. We demonstrate the utility of our metric by evaluating it for various traditional disentanglement methods, as well as for LSBD-specific methods (including our own). From these experiments we highlight three results: (1) traditional VAEbased disentanglement methods do not learn LSBD representations, (2) LSBDThe contents of this chapter are largely based on our paper Quantifying and Learning Linear Symmetry-Based Disentanglement (Tonnaer et al., 2022), although the DLSBDmetric presented in Section 4.4 was first introduced in our paper A Metric for Linear Symmetry-Based Disentanglement (Pérez Rey et al., 2020). We applied the LSBD-VAE method from Section 4.5 in the SHREC 2021 3D Object Retrieval Challenge (Sipiran et al., 2021), the results of this are described in Section 4.8.
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