64 Quantifying and Learning Linear Symmetry-Based Disentanglement (LSBD) disentangled and accordingly splits into irreducible representations ρk : G→Zk where Z =Z1 ⊕· · ·⊕ZK for some K∈N. We will define a new inner product ⟨·, ·⟩ρ,h,µ onZ as follows. First of all, we declare Zk andZk′ to be orthogonal with respect to ⟨·, ·⟩ρ,h,µ whenever k̸ =k′. We denote the orthogonal projection onZk by πk. For z,z′ ∈Zk, we set ⟨z,z′⟩ρ,h,µ :=λ−1 k,h,µZ g∈G (ρ(g) · z,ρ(g) · z′)dm(g), (4.11) where mis the (bi-invariant) Haar measure normalised such that m(G)=1, and λk,h,µ :=Z XZG∥ πk(h(x))∥ 2dm(g)dµ(x) (4.12) if the integral on the right-hand side is strictly positive and otherwise we set λk,h,µ :=1. This construction completely specifies the new inner product and has the following properties: • The subspaces Zk are mutually orthogonal. • ρk(g) is orthogonal onZk for every g ∈G. In other words, ρk maps to the orthogonal group onZk. Moreover, ρ maps to the orthogonal group onZ. This follows directly from the bi-invariance of the Haar measure and the definition of ⟨·, ·⟩ρ,h,µ. • If πk is the orthogonal projection to Zk, then ZX∥ πk(h(x))∥ 2 ρ,h,µdµ(x)=1 (4.13) if the integral on the left is strictly positive. For an arbitrary pair z,z′ ∈Z the inner product ⟨·, ·⟩ρ,h,µ is given by ⟨z,z′⟩ρ,h,µ = KX k=1 λ−1 k,h,µZ g∈G (ρ(g) · πk(z),ρ(g) · πk(z′))dm(g). (4.14) This inner product naturally induces the norm∥ · ∥ρ,h,µ for any z ∈Z as ∥z∥ρ,h,µ =q⟨z,z⟩ρ,h,µ. (4.15)
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