4.4 DLSBD: Quantifying LSBD 67 Figure 4.2: Intuitive description of the practical computation of DLSBD. Consider a dataset modelled by a group decomposition G=G1 ×· · ·×GK acting on x0 and embedded in a latent space Z via h. In this example, the subgroup Gk = SO(2) models the rotations of an airplane. Other subgroups G̸=k could also have some action, e.g. changes in airplane colour. The first step to calculate the disentanglement of Gk is to construct a set of data encodings Zk ⊆Z whose variability is due to Gk. These encodings are then projected into a 2-dimensional space through PCA. For these projected encodings we can describe the group representations in a simple parametric formρk,w. For a givenρk,w, the equivariance of Gk is measured as the dispersion after applying the action of the inverse group representationρ−1 k,w. Similar to Cohen and Welling (2014), we find a suitable change of basis that exposes the invariant subspace Zk corresponding to the k-th subgroup Gk. The new basis is obtained from the eigenvectors resulting from applying Principal Component Analysis (PCA) to Zk. Each element inZk is projected into the first Dk eigenvectors. The new set is denoted as Z′k ⊆RDk with elements z′k(g1, . . . ,gK) ⊆RDk that are the projected versions of zk(g1, . . . ,gK). Quessard et al. (2020) describe how one could parameterise the subgroup representations of SO(Dk) for arbitrary Dk, but here we focus onGk =SO(2). In this case, we can parameterise each subgroup representation in terms of a single integer parameter ω ∈ Z as ρk,ω(gk) corresponding to a 2 ×2 rotation matrix whose angle of rotation is ω multiplied by the known angle associated to the group element gk ∈Gk =SO(2). For this subgroup we can approximate the Mρ,h,x0 in Definition 4.3 as Mk,ω given by Mk,ω := 1 |G| X (g1,...,gK)∈G ρk,ω(g−1 k ) · z′(g1, . . . ,gK). (4.22) Similar to Definition 4.3, we would like to find the optimal ρk,ω that minimises the integral over the group representations. We can define a parameter search
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