In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation , let d , m , n , r ∈ N d,m,n,r\in\mathbb{N} d , m , n , r ∈ N 2 d = d + d 2d=d+d 2 d = d + d L 2 L^{2} L 2 Σ k 2 \Sigma^{2}_{k} Σ k 2 k ∈ N k\in\mathbb{N} k ∈ N W ^ 2 \widehat{W}_{2} W 2 κ k \kappa_{k} κ k T # T_{\#} T # m i \mathrm{m}_{i} m i m i j \mathrm{m}_{ij} m ij M ^ \widehat{M} M L 2 L^{2} L 2 Π 2 \Pi^{2} Π 2 I \mathcal{I} I Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost ; affine data, their norms ∥ T ∥ \lVert T\rVert ∥ T ∥ p r 1 , p r 2 , D , d i a g \mathrm{pr}^{1},\mathrm{pr}^{2},D,\mathrm{diag} pr 1 , pr 2 , D , diag Affine Data and Affine Substitutions of Noncommutative Polynomials ; continuity of maps between metric spaces is continuity on the whole domain , R \mathbb{R} R absolute-value metric . Let T = ( A , c ) T=(A,c) T = ( A , c ) m m m n n n S S S n n n r r r
1. (Bounded laws) ¶ For λ ∈ Σ m \lambda\in\Sigma_{m} λ ∈ Σ m i , j ∈ [ m ] i,j\in[m] i , j ∈ [ m ]
T # κ m ( λ ) = κ n ( λ ∘ σ T ) , m i ( κ m ( λ ) ) = λ ( x i ) , m i j ( κ m ( λ ) ) = λ ( x i x j ) , M ^ ( κ m ( λ ) ) = M ( λ ) . T_{\#}\kappa_{m}(\lambda)=\kappa_{n}(\lambda\circ\sigma_{T}),\qquad\mathrm{m}_{i}(\kappa_{m}(\lambda))=\lambda(x_{i}),\qquad\mathrm{m}_{ij}(\kappa_{m}(\lambda))=\lambda(x_{i}x_{j}),\qquad\widehat{M}(\kappa_{m}(\lambda))=M(\lambda). T # κ m ( λ ) = κ n ( λ ∘ σ T ) , m i ( κ m ( λ )) = λ ( x i ) , m ij ( κ m ( λ )) = λ ( x i x j ) , M ( κ m ( λ )) = M ( λ ) . For μ , ν ∈ Σ d \mu,\nu\in\Sigma_{d} μ , ν ∈ Σ d γ ∈ Σ 2 d \gamma\in\Sigma_{2d} γ ∈ Σ 2 d κ 2 d ( γ ) ∈ Π 2 ( κ d ( μ ) , κ d ( ν ) ) \kappa_{2d}(\gamma)\in\Pi^{2}(\kappa_{d}(\mu),\kappa_{d}(\nu)) κ 2 d ( γ ) ∈ Π 2 ( κ d ( μ ) , κ d ( ν )) γ ∈ Π ( μ , ν ) \gamma\in\Pi(\mu,\nu) γ ∈ Π ( μ , ν ) I ( κ 2 d ( γ ) ) = I ( γ ) \mathcal{I}(\kappa_{2d}(\gamma))=I(\gamma) I ( κ 2 d ( γ )) = I ( γ )
2. (Lipschitz estimates) ¶ For all μ , ν ∈ Σ m 2 \mu,\nu\in\Sigma^{2}_{m} μ , ν ∈ Σ m 2 i , j ∈ [ m ] i,j\in[m] i , j ∈ [ m ] M ^ ( μ ) 1 / 2 \widehat{M}(\mu)^{1/2} M ( μ ) 1/2 M ^ ( μ ) ≥ 0 \widehat{M}(\mu)\ge0 M ( μ ) ≥ 0
W ^ 2 ( T # μ , T # ν ) ≤ ∥ T ∥ W ^ 2 ( μ , ν ) , ∣ m i ( μ ) − m i ( ν ) ∣ ≤ W ^ 2 ( μ , ν ) , ∣ M ^ ( μ ) 1 / 2 − M ^ ( ν ) 1 / 2 ∣ ≤ W ^ 2 ( μ , ν ) , \widehat{W}_{2}(T_{\#}\mu,T_{\#}\nu)\le\lVert T\rVert\,\widehat{W}_{2}(\mu,\nu),\qquad|\mathrm{m}_{i}(\mu)-\mathrm{m}_{i}(\nu)|\le\widehat{W}_{2}(\mu,\nu),\qquad\bigl|\widehat{M}(\mu)^{1/2}-\widehat{M}(\nu)^{1/2}\bigr|\le\widehat{W}_{2}(\mu,\nu), W 2 ( T # μ , T # ν ) ≤ ∥ T ∥ W 2 ( μ , ν ) , ∣ m i ( μ ) − m i ( ν ) ∣ ≤ W 2 ( μ , ν ) , M ( μ ) 1/2 − M ( ν ) 1/2 ≤ W 2 ( μ , ν ) , ∣ m i j ( μ ) − m i j ( ν ) ∣ ≤ W ^ 2 ( μ , ν ) ( M ^ ( μ ) 1 / 2 + M ^ ( ν ) 1 / 2 ) . |\mathrm{m}_{ij}(\mu)-\mathrm{m}_{ij}(\nu)|\le\widehat{W}_{2}(\mu,\nu)\bigl(\widehat{M}(\mu)^{1/2}+\widehat{M}(\nu)^{1/2}\bigr). ∣ m ij ( μ ) − m ij ( ν ) ∣ ≤ W 2 ( μ , ν ) ( M ( μ ) 1/2 + M ( ν ) 1/2 ) . In particular T # T_{\#} T # m i \mathrm{m}_{i} m i m i j \mathrm{m}_{ij} m ij M ^ \widehat{M} M
3. (Functoriality) ¶ ( S ∘ T ) # = S # ∘ T # (S\circ T)_{\#}=S_{\#}\circ T_{\#} ( S ∘ T ) # = S # ∘ T # ( i d m ) # (\mathrm{id}_{m})_{\#} ( id m ) # Σ m 2 \Sigma^{2}_{m} Σ m 2
4. (Moments) ¶ Let μ ∈ Σ m 2 \mu\in\Sigma^{2}_{m} μ ∈ Σ m 2 i , j ∈ [ m ] i,j\in[m] i , j ∈ [ m ] ξ 1 , … , ξ m \xi_{1},\dots,\xi_{m} ξ 1 , … , ξ m
m i j ( μ ) = m j i ( μ ) , m i ( μ ) 2 ≤ m i i ( μ ) , ∑ i = 1 m ∑ j = 1 m ξ i ξ j m i j ( μ ) ≥ 0 , M ^ ( μ ) ≥ 0 , \mathrm{m}_{ij}(\mu)=\mathrm{m}_{ji}(\mu),\qquad\mathrm{m}_{i}(\mu)^{2}\le\mathrm{m}_{ii}(\mu),\qquad\sum_{i=1}^{m}\sum_{j=1}^{m}\xi_{i}\xi_{j}\,\mathrm{m}_{ij}(\mu)\ge0,\qquad\widehat{M}(\mu)\ge0, m ij ( μ ) = m ji ( μ ) , m i ( μ ) 2 ≤ m ii ( μ ) , i = 1 ∑ m j = 1 ∑ m ξ i ξ j m ij ( μ ) ≥ 0 , M ( μ ) ≥ 0 , and for all i , k ∈ [ n ] i,k\in[n] i , k ∈ [ n ]
m i ( T # μ ) = c i + ∑ j = 1 m A i j m j ( μ ) , \mathrm{m}_{i}(T_{\#}\mu)=c_{i}+\sum_{j=1}^{m}A_{ij}\,\mathrm{m}_{j}(\mu), m i ( T # μ ) = c i + j = 1 ∑ m A ij m j ( μ ) , m i k ( T # μ ) = c i c k + c i ∑ l = 1 m A k l m l ( μ ) + c k ∑ j = 1 m A i j m j ( μ ) + ∑ j = 1 m ∑ l = 1 m A i j A k l m j l ( μ ) . \mathrm{m}_{ik}(T_{\#}\mu)=c_{i}c_{k}+c_{i}\sum_{l=1}^{m}A_{kl}\,\mathrm{m}_{l}(\mu)+c_{k}\sum_{j=1}^{m}A_{ij}\,\mathrm{m}_{j}(\mu)+\sum_{j=1}^{m}\sum_{l=1}^{m}A_{ij}A_{kl}\,\mathrm{m}_{jl}(\mu). m ik ( T # μ ) = c i c k + c i l = 1 ∑ m A k l m l ( μ ) + c k j = 1 ∑ m A ij m j ( μ ) + j = 1 ∑ m l = 1 ∑ m A ij A k l m j l ( μ ) . 5. (Cost) ¶ For every γ ∈ Σ 2 d 2 \gamma\in\Sigma^{2}_{2d} γ ∈ Σ 2 d 2 γ ∈ Π 2 ( p r # 1 γ , p r # 2 γ ) \gamma\in\Pi^{2}(\mathrm{pr}^{1}_{\#}\gamma,\mathrm{pr}^{2}_{\#}\gamma) γ ∈ Π 2 ( pr # 1 γ , pr # 2 γ ) I ( γ ) ≥ 0 \mathcal{I}(\gamma)\ge0 I ( γ ) ≥ 0
I ( γ ) = M ^ ( p r # 1 γ ) + M ^ ( p r # 2 γ ) − 2 ∑ j = 1 d m j , d + j ( γ ) , W ^ 2 ( p r # 1 γ , p r # 2 γ ) 2 ≤ I ( γ ) . \mathcal{I}(\gamma)=\widehat{M}(\mathrm{pr}^{1}_{\#}\gamma)+\widehat{M}(\mathrm{pr}^{2}_{\#}\gamma)-2\sum_{j=1}^{d}\mathrm{m}_{j,d+j}(\gamma),\qquad\widehat{W}_{2}(\mathrm{pr}^{1}_{\#}\gamma,\mathrm{pr}^{2}_{\#}\gamma)^{2}\le\mathcal{I}(\gamma). I ( γ ) = M ( pr # 1 γ ) + M ( pr # 2 γ ) − 2 j = 1 ∑ d m j , d + j ( γ ) , W 2 ( pr # 1 γ , pr # 2 γ ) 2 ≤ I ( γ ) . In particular W ^ 2 ( μ , ν ) 2 ≤ I ( γ ) \widehat{W}_{2}(\mu,\nu)^{2}\le\mathcal{I}(\gamma) W 2 ( μ , ν ) 2 ≤ I ( γ ) μ , ν ∈ Σ d 2 \mu,\nu\in\Sigma^{2}_{d} μ , ν ∈ Σ d 2 γ ∈ Π 2 ( μ , ν ) \gamma\in\Pi^{2}(\mu,\nu) γ ∈ Π 2 ( μ , ν )
6. (Diagonal coupling) ¶ For every μ ∈ Σ d 2 \mu\in\Sigma^{2}_{d} μ ∈ Σ d 2 d i a g # μ ∈ Π 2 ( μ , μ ) \mathrm{diag}_{\#}\mu\in\Pi^{2}(\mu,\mu) diag # μ ∈ Π 2 ( μ , μ ) I ( d i a g # μ ) = 0 \mathcal{I}(\mathrm{diag}_{\#}\mu)=0 I ( diag # μ ) = 0