60 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) grid, where the edges of the grid wrap around. They use the knowledge that irreducible group representations for this scenario take the form of 2D rotation matrices. Thus, although this indeed measures important aspects of the LSBD definition, it is not a general quantification of LSBD and can only be used in this specific case. Their second metric, Factor Leakage, only measures the number of dimensions in which the subgroup actions are encoded, which is not a property required by LSBD. It is derived from the Mutual Information Gap (MIG) (Chen et al., 2018), which is a traditional disentanglement metric that measures the difference in information content between the first and second most informative latent dimension. This favours using a single dimension to model the variability due to a particular subgroup, but in general LSBD representations frequently need two or more dimensions as was already shown by Higgins et al. (2018). Nevertheless, the LSBD definition does not require the vector subspace corresponding to a particular subgroup to be axis-aligned, so this is not a suitable nor general quantification of LSBD either. Alternative symmetry-based disentanglement definition Bouchacourt et al. (2021) propose an alternative definition of disentanglement that is similar to LSBD but with one key difference. Specifically, they omit the requirement that the latent space should decompose into independent subspaces, acted on by a single subgroup. Their definition still considers a symmetry groupG that decomposes into subgroups G1, . . . ,GK and requires the encoding function h : X→Z to be equivariant w.r.t. the actions of these subgroups onXandZ, but each subgroup is allowed to act on the entire latent space. They refer to these subgroup actions as distributed operators. This more flexible definition can handle non-commutative subgroups as well, unlike the LSBD definition of Higgins et al. (2018). In particular, this view of disentanglement can prevent needing to introduce topological defects (i.e. discontinuities in the encoder). In this thesis, we restrict ourselves to LSBD, dealing with topological effects by imposing suitable topologies on the latent space, but this alternative view provides an interesting way forward towards more complicated group decompositions, in particular with non-commutative subgroups.
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