4.4 DLSBD: Quantifying LSBD 65 Alternative view of DLSBD We now present an alternative expression for the disentanglement metric DLSBD. This expression relates more visibly to the definition of equivariance. Let ρ ∈P(G,Z) be a linear disentangled representation of GinZ. By expanding the inner product (or by using common computation rules for expectations and variances), we first find that ZG ρ(g)−1 · h(g · x0) −Z G ρ(g′)−1 · h(g′ · x0)dν(g′) 2 ρ,h,µ dν(g) =Z G ρ(g)−1 · h(g · x0) 2 ρ,h,µ dν(g) − Z G ρ(g)−1 · h(g · x0)dν(g) 2 ρ,h,µ = 1 2 ZGZG∥ ρ(g)−1 · h(g · x0) −ρ(g′)−1 · h(g′ · x0)∥ 2 ρ,h,µdν(g)dν(g′). (4.16) Using the fact that ρ maps to the orthogonal group for ⟨·, ·⟩ρ,h,µ, we can write the same expression as 1 2 ZGZG∥ ρ(g ◦g′−1)−1 · h(((g ◦g′−1) ◦g′) · x0) −h(g′ · x0)∥ 2 ρ,h,µdν(g)dν(g′). (4.17) This brings us to the alternative characterisation of DLSBD as DLSBD = inf ρ∈P(G,Z) 1 2 ZGZG∥ ρ(g ◦g′−1)−1h(((g ◦g′−1) · g′) · x0) −h(g′ · x0)∥ 2 ρ,h,µdν(g)dν(g′). (4.18) In particular, if for every data point xthere is a unique group element gx such that x=gx · x0, the disentanglement metric DLSBD can also be written as DLSBD = inf ρ∈P(G,Z) 1 2 ZGZX∥ ρ(g ◦g−1 x )−1h((g ◦g−1 x ) · x) −h(x)∥ 2 ρ,h,µdν(g)dµ(x), (4.19) in which the equivariance condition appears prominently. The condition becomes even more apparent if ν is in fact the Haar measure itself, in which case the
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