4.2 LSBD Definition 55 define · : G×X→Xsuch that g · x=b g · b−1(x) . (4.2) Here, b−1(x) exists because b is injective2. Note that b−1(x) ∈W, thus the group action on the right-hand side of Equation (4.2) is the one onW. Given this action of Z onX, we can conclude that if his equivariant (with respect to the actions of Gon Xand Z), it follows that f is equivariant (with respect to the actions of GonWandZ). To see this, let hbe equivariant, i.e. g · h(x)=h(g · x) ∀g ∈G,x∈X. (4.3) From Equation (4.2) it follows that g · h(x)=h b g · b−1(x) =f g · b−1(x) . (4.4) From the fact that h(x)=h(b(b−1(x)))=f(b−1(x)) and substitutingw=b−1(x), it follows that Equation (4.1) holds, i.e. f is equivariant. Symmetry-based disentanglement definitions We can now proceed to define disentanglement in terms of the encoding function h. Given the assumption described above that b is injective, the definition here corresponds to the original definition given in terms of the function f (Higgins et al., 2018). This definition formalises the requirement that the decomposed symmetries of the world should act on specific subsets of a learned representation. Definition 4.1 (SBD: Symmetry-Based Disentanglement) A model’s encoding function h : X →Z is SBD with respect to the group decomposition G=G1 × . . . ×GK if the following conditions hold: 1. There is a decomposition of the embedding space Z =Z1 ×. . . ×ZK into K subspaces, 2. There are group actions for each subgroup Gk on the corresponding subspace Zk, i.e. ·k : Gk ×Zk →Zk for k =1, . . . ,K. 2Technically, b−1 is only defined for b(W), i.e. the image of Wunder b, which may in theory be a proper subset of X. However, since we assume that any data observations are the result of the generative process b, we are only interested in the group action on b(W). For simplicity and readability however, we write this as the action onX.
RkJQdWJsaXNoZXIy MjY0ODMw