Optimal transport with invariances between Gaussian mixture models Antoine Salmona, Julie Delon, Agnès Desolneux Humboldt University of Berlin, OT Workshop, March 2024

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GMMd (K) = {⋃ } π,m,Σ Gaussian mixture models [Xia et al, Neurips 2022] Image completion Semantic segmentation [Liang et al, Neurips 2022] Rs2 µ1 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GMM on patches Style, texture editing, restoration [Leclaire, Rabin 2019] [EPLL 2011, SPLE 2013, HDMI 2018] Autoencoders, GMM on latent space K ∑ k=1 πk 𝒩 (mk , Σk ) GMMd (∞) = ⋃ K≥1 GMMd (K) [Saseendran et al., Neurips 2021]

Wasserstein Distance , de fi nes a distance between probability measures. c(x, y) = d(x, y)p with p ≥ 1 and d a distance Wp (μ0 , μ1 ) = ( inf γ∈Π(μ0 ,μ1 ) ∬ c(x, y)dγ(x, y)) 1 p = inf (X,Y)∼(μ0 ,μ1 ) [ 𝔼 [dp(X, Y)]|] 1/p μ0 μ1 c(x, y) x y p=2 or 1 often used in applications Wasserstein distances and barycenters [Solomon et al. 2015] Wasserstein Barycenter of for weights (νi )i∈{1,…,p} ∑ i λi = 1 ν* ∈ argmin ρ p ∑ i=1 λi W2 2 (νi , ρ) [Agueh, Carlier 2011]: existence and unicity of if the vanish on small sets. ν* νi

OT between Gaussians, quadratic cost Gaussian distributions on μi = 𝒩 (mi , Σi ), i ∈ {0,1,…, I − 1} ℝd W2 2 (μ0 , μ1 ) = ∥m0 − m1 ∥2 + tr ( Σ0 + Σ1 − 2 (Σ1 2 0 Σ1 Σ1 2 0) 1 2 ) B2(Σ0 ,Σ1 ) A ffi ne optimal map T(x) = m1 + Σ−1 2 0 (Σ1 2 0 Σ1 Σ1 2 0) 1 2 Σ− 1 2 0 (x − m0 ) Barycenter [Agueh, Carlier 2011]: , 𝒩 (m*, Σ*) = argmin ρ I−1 ∑ i=1 λi W2 2 (μi , ρ) m* = ∑ λi mi Σ* = min Σ ∑ i λi B2(Σ, Σi ) [Dowson, Landau, 82] ,[Givens, Shortt 84]… Texture mixing [Xia et al, 2014 ] [Vacher et al., 2020] FID [Heusel et al., 2017]

What about GMMs?

Optimal transport between GMM? µ0 = N(0, 1) µ1 = 1 2 ( 1 + 1) 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 ft(x) = 1 1 t ✓ g ✓ x + t 1 t ◆ 1x< t + g ✓ x t 1 t ◆ 1x>t ◆ 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 density of µt : 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 • • • μ0 = 𝒩 (0,1) μ1 = 1 2 𝒩 (−5,0.1) + 1 2 𝒩 (5,0.1) OT plans / barycenters between GMM are usually not GMM themselves

Prop. [D.,D.] metric on Geodesic space. MW2 GMMd (∞) . Optimal transport between GMM? Def. [D., Desolneux 2019] MW2 2 (μ0 , μ1 ) := inf γ∈Π(μ0 ,μ1 )∩GMM2d (∞) ∫ ℝd×ℝd ∥y0 − y1 ∥2dγ(y0 , y1 ) Prop. [D.,D.] MW2 2(μ0 , μ1 ) = min w∈Π(π0 ,π1 ) ∑ k,l wkl W2 2 (μk 0 , μl 1 ) . OT problem K0 × K1 See also [Chen, Georgiu, Tannenbaum, 2017], [Chen, Ye, Li, 2016], [Lambert et al. 2023], etc , , on μ0 = K0 ∑ i=1 πi 0 μi 0 𝒩 (mi 0 , Σi 0 ) μ1 = K1 ∑ i=1 πi 1 μi 1 𝒩 (mi 1 , Σi 1 ) ℝd μ0 0 μ1 0 μ0 1 μ1 1 μ2 1

Optimal plan and barycenters At least one solution has less than components! K0 + K1 − 1 Optimal plan for = solution of the OT pb. = optimal map between and MW2 γ*(x, y) = ∑ k,l w* k,l pμk 0 (x) δy=Tk,l (x) w* K0 × K1 Tkl μk 0 μl 1 Barycenters for GMM solution with less than components MW2 K0 + K1 + … + KI−1 − I + 1 MW2 W2

Examples of applications Color transfer Texture synthesis [with A. Leclaire, 2022] Evaluating generative models (FID) [Luzi et al. 2022]

Invariance? Spaces of dimensions? ≠

Gromov Wasserstein between incomparable spaces Discrete case cost matrix cost matrix μ0 = ∑ i πi 0 δxi Cx μ1 = ∑ j πj 1 δyj Cy GWp p (Cx, Cy, μ0 , μ1 ) := min w∈Π(π0 ,π1 ) ∑ i,j,k,l Cx i,k − Cy j,l p wi,j wk,l Cx i,k Cy j,l i k j l Non convex quadratic assignment problem, NP-hard Approximate algos required. GWp p (μ0 , μ1 ) := inf γ∈Π(μ0 ,μ1 ) ∫ 𝒳 × 𝒴 ∫ 𝒳 × 𝒴 |c 𝒳 (x, x′  ) − c 𝒴 (y, y′  )|p dγ(x, y)dγ(x′  , y′  ) [Mémoli 2011, Sturm 2012, Solomon et al. 2016, Vayer et al. 2020 etc…] If and , , pseudo-metric between metric measure spaces, i.e. i ff isometry s.t. c 𝒳 = dq 0 c 𝒴 = dq 1 q ≥ 1 GWp GWp (μ0 , μ1 ) = 0 ∃ψ : ( 𝒳 , d0 ) → ( 𝒴 , d1 ) μ1 = ψ#μ0 w?

Gromov-W between Gaussians Prop. [Salmona, D., Desolneux, 22] If , , GW admits Gaussian opt. plans of the form , with , and c 𝒳 (x, x′  ) = ⟨x, x′  ⟩d c 𝒴 (y, y′  ) = ⟨y, y′  ⟩d′  γ* = (Idd′  , T)#μ0 T(x) = m1 + P1 AP0 T(x−m0 ) A = ( ˜ Idd′  D1 1 2 D(d′  ) 0 −1 2 ) [d′  ,d] , with ( eigenvalues) on μ0 = 𝒩 (m0 , Σ0 ) Σ0 = P0 D0 PT 0 ↘ ℝd , with ( eigenvalues) on μ1 = 𝒩 (m1 , Σ1 ) Σ1 = P1 D1 PT 1 ↘ ℝd′  d ≥ d′  Classical OT Gromov [included in POT by Rémi Flamary]

GW between Gaussians Prop. [Salmona et al. 22] If and , same solution by restricting the optimization to Gaussian couplings of . c 𝒳 (x, x′  ) = ∥x − x′  ∥2 c 𝒴 (y, y′  ) = ∥y − y′  ∥2 (μ0 , μ1 ) Generalizes a result by [Vayer et al. 20] on linear Gromov-Monge maps. Writing we can show that , where GGWp p (μ0 , μ1 ) := inf γ∈Π(μ0 ,μ1 )∩ 𝒩 (ℝd+d′  ) ∫ ∫ |∥x − x′  ∥2 − ∥y − y′  ∥2|2 dγ(x, y)dγ(x′  , y′  ) LGW2 (μ, ν) ≤ GW2 (μ, ν) ≤ GGW2 (μ, ν) GGW2 (μ, ν) = 4(tr(D0 ) − tr(D1 ))2 + 8tr ((D(d′  ) 0 − D1 )2 ) + 8 (tr (D2 0 ) − tr ((D(d′  ) 0 )2 )) LGW2 (μ, ν) = 4(tr(D0 ) − tr(D1 ))2 + 4tr ((D(d′  ) 0 − D1 )2 ) + 4 (tr (D2 0 ) − tr ((D(d′  ) 0 )2 )) +4( tr (D2 1 ) − tr (D2 0 ))2

Between GMMs? Gromovization of MW2 on μ = K ∑ k=1 πkμk ℝd on ˜ μ = K ∑ k=1 πkδμk 𝒩 (ℝd) [Lambert et al. 2023] MW2 2(μ0 , μ1 ) = min w∈Π(π0 ,π1 ) ∑ k,l wkl W2 2 (μk 0 , μl 1 ) . OT problem between two discrete measures on 𝒩 (ℝd) MGW2 2 (μ0 , μ1 ) := min w∈Π(π0 ,π1 ) ∑ i,j,k,l W2 2 (μi 0 , μk 0 ) − W2 2 (μj 1 , μl 1 ) 2 wi,j wk,l GW2 (W2 2 ,W2 2 , ˜ μ0 , ˜ μ1 )

Between GMMs? Gromovization of MW2 on μ0 = K0 ∑ i=1 πi 0 𝒩 (mi 0 , Σi 0 ) ℝd on μ1 = K1 ∑ i=1 πi 1 𝒩 (mi 1 , Σi 1 ) ℝd′  Def. MGW2 2 (μ0 , μ1 ) := min w∈Π(π0 ,π1 ) ∑ i,j,k,l W2 2 (μi 0 , μk 0 ) − W2 2 (μj 1 , μl 1 ) 2 wi,j wk,l • Pseudo-metric on the set of GMM of arbitrary fi nite dimensions. • If with isometry for the Euclidean norm, then (but not only). • No straightforward equivalence with a continuous formulation by restricting the couplings to GMM. μ1 = T#μ0 T MGW2 2 (μ0 , μ1 ) = 0

MGW2 Galloping horse experiment, originally conducted in (Rustamov et al., 2013), (Solomon et al., 2016), (Vayer et al., 2019). Computational GW : Fit GMMs on data and compute dist. between the two GMMs Computational cost of the method ≈ fi tting the GMMs MGW2 Computational GW: • Entropic GW [Peyré et al., 2016, Solomon et al., 2016] • Sliced GW [Vayer et al., 2019] • Minibatch GW [Fatras et al., 2021] • Low-Rank GW [Scetbon et al., 2022] • Quantized GW [Chowdhury et al., 2022]…

in practice MGW2 No obvious way to derive a transport plan. If d ≥ d′  , inf ϕ∈Isom d′  (ℝd) ∑ k,l w* k,l W2 (μk , ϕ#νl ) [Mazur, Ulam 32] i ff with , Stiefel manifold ϕ ∈ Isomd′  (ℝd) ϕ(x) = Px + b P ∈ 𝕍 d′  (ℝd), b ∈ ℝd 𝕍 d′  (ℝd) = {P ∈ ℝd×d′  , PTP = Idd′  } = inf P∈ 𝕍 d′  (ℝd), b∈ℝd ∑ k,l w* k,l W2 (μk , (P. + b)#νl ) Transport Plan with the OT map between and γ*(x, y) = ∑ k,l w* k,l pμk (x)δy=Tk,l(P*T(x−b*)) Tk,l μk (P*. + b*)#νl optimal discrete plan given by . w* GMW2

Applications MGW2 MGW2 MGW2

Questions?