56 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) 3. The group Gacts onZ as g · z =(g1 ·1 z1, . . . ,gK·KzK), for g = (g1, . . . ,gK) ∈ Gand z = (z1, . . . ,zK) ∈ Z, with gk ∈ Gk and zk ∈Zk for k =1, . . . ,K. In other words, each subspace Zk is only affected by the action of subgroup Gk, and remains fixed by the actions of all other subgroups Gj with j̸ =k. 4. The function h is equivariant with respect to the actions of Gon Xand Z, i.e. for all x∈Xand g ∈Git holds that h(g · x)=g · h(x). It is desirable to impose an additional constraint on the form that the group action of Gon Z takes, namely that this action transforms the corresponding disentangled subspace linearly. This gives a certain regularity to the action and the encodings in Z. In particular, any transformation g ∈ Gcan be described with a single transformation matrix (given a suitable basis for Z), which gives a simple descriptor of how encodings are transformed by the action. Furthermore, this imposes a certain structure or topology onZ, yielding encodings that reflect the symmetries of the world well. Therefore, we also define Linear SBD (LSBD), with the additional constraint that the action of GonZ is linear. In particular, this means that the action can be formulated as a group representation3 ρ : G→GL(Z) onZ(see Section 2.3). In this case, Z must be a vector space, and we write the decomposition of Z as a direct sumZ1 ⊕. . . ⊕ZK. In this thesis we focus specifically on LSBD, which is defined as follows. Definition 4.2 (LSBD: Linear Symmetry-Based Disentanglement) A model’s encoding function h : X→Z, where Z is a vector space, is LSBD with respect to the group decompositionG=G1 ×. . . ×GK if the following conditions hold: 1. There is a decomposition of the embedding space Z =Z1 ⊕. . . ⊕ZK into K vector subspaces, 2. There are group representations ρk for each subgroupGk on the corresponding subspace Zk, i.e. ρk : Gk →GL(Zk) for k =1, . . . ,K. 3Note the double meaning of the wordrepresentation. Agroup representation refers to a linear group action. A learned/latent representation is the result of an encoding function h : X →Z, which we will sometimes refer to as anencoding aswell.
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