5.2 Related Work 109 Symmetry-Based Disentanglement Disentanglement is typically explained from the perspective of independent factors of variation, but this doesn’t provide a formal definition of what constitutes a disentangled representation. To address this, Symmetry-Based Disentanglement (SBD) (Higgins et al., 2018) gives a formal definition by means of group theory. The motivation is that symmetries in the real world are what leads to variability in data observations. A symmetry is a transformation that affects some factor of variation, but leaves all others invariant. Such symmetry transformations can be described with the formal language of group theory, as a decomposable transformation group acting on both the data and the learned representations. The goal of SBD is then to learn representations that are invariant to such symmetries, resulting in disentangled subspaces where one symmetry (or subgroup) affects only one subspace. Several recent works focus on learning SBD representations (Caselles-Dupré et al., 2019; Painter et al., 2020; Quessard et al., 2020; Tonnaer et al., 2022), with varying degrees of supervision on transformations. We focus on Linear SBD (LSBD), where an additional requirement is that transformations act on the representations as linear operators (i.e. matrix multiplications). This allows for a simple and consistent description of transformations in latent space, which is useful for generalising beyond factor combinations seen during training. The exact definitions of SBD and LSBD are given in Section 4.2. OOD Generalisation Generalisation has always been one of the main goals in machine learning, and representation learning in particular. Machine learning methods are however mostly based on the i.i.d. assumption that train and test data are independent and identically distributed. In realistic scenarios this is hard to realise, so there is significant interest in dropping this assumption and investigating out-ofdistribution (OOD) generalisation, see e.g. Shen et al. (2021) for an overview. Disentanglement seems a sensible solution to improve OOD generalisation, especially from the perspective of combinatorial generalisation where the goal is to generalise to unseen combinations of factor values observed during training. However, Montero et al. (2021) show somewhat surprisingly that disentanglement does not seem to help with combinatorial generalisation. In our work, we follow their evaluation protocol and expand on it, focusing mostly on LSBD instead of traditional disentanglement. Our setting is similar to the “composition” setting by Schott et al. (2022), which provides a benchmark to evaluate OOD detection for visual representation learning. Related to this, Träuble et al.
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