4.4 DLSBD: Quantifying LSBD 63 of any encoding function h : X →Z given a data probability measure µ on X, provided that µ can be written as the pushforward GX(·,x0)#ν of some probability measure ν on Gby the function GX(·,x0) for some base point x0. More formally, µ(A)=GX(·,x0)#ν(A) =ν({g ∈G| GX(g,x0) ∈A}), (4.8) for Borel subsets A ⊂ X. Note that this is only possible if the action GX is transitive, i.e. for all x,y ∈X(in particular x =x0) there exists a g ∈Gsuch that GX(g,x)=g · x=y. For example, the situation of a dataset withNdata points {xn} N n=1 ={gn · x0} N n=1 corresponds to the case in whichν andµare empirical measures on the group Gand data space X, respectively: ν := 1 N NX i=1 δgi, µ:= 1 N NX i=1 δxi. (4.9) In particular, if we assume that each data point xn is unique, this means that we assume that the action of GonXis also free, which together with the fact that the action is transitive means that the action is regular. Definition 4.3 (DLSBD) Given a group Gwitha transitive actionGX, a probability measure ν on G, a data measure µ=GX(·,x0)#ν for some base point x0 ∈X, and an encoding functionh: X→Z, we define the metric DLSBD as DLSBD := inf ρ∈P(G,Z) ZG ρ(g)−1 · h(g · x0) −Mρ,h,x0 2 ρ,h,µ dν(g), with Mρ,h,x0 =Z G ρ(g′)−1 · h(g′ · x0)dν(g′), (4.10) where the norm∥ · ∥ρ,h,µ is a Hilbert-space norm depending on the representation ρ, the encoding function h : X →Z, and the data measure µ. More details of this norm are described below. Moreover, P(G,Z) denotes the set of linearly disentangled representations of Gin Z. Lower values of DLSBD indicate better disentanglement, zero being optimal. Description of the norm∥ · ∥ρ,h,µ from an inner product To describe the norm∥ · ∥ρ,h,µ used in Definition 4.3, we start with an arbitrary inner product (·, ·) on the linear latent space Z. Assume that ρ is linearly
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