Proof of Wasserstein Estimates for Affine Push-Forwards, First and Quadratic Moments, and the Cost of a Joint Law

Each result cited below is universally quantified over the data in its own statement. Finite sums follow the rules of Properties of Finite Sums of Vectors and Properties of Finite Sums: claim 1 gives the recursion, claim 2 additivity, claim 3 homogeneity, claim 4 (vectors) says a linear map commutes with a finite sum, claim 6 (numbers) bounds a nonnegative summand by the sum, and claim 7 treats a single possibly nonzero summand. Claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers compares finite sums of real numbers termwise. For $k\in\mathbb{N}$ and a tracial state $\lambda$ on $\mathcal{P}_{k}$ we use the norm $\|p\|_{\lambda}$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws. We also use the pairing $\beta_{\lambda}(a,b)=\lambda(ab)$ on the real vector space $\mathcal{P}_{k,\mathrm{sa}}$. By Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §pairing (for $\lambda$) it is real-valued, symmetric, bilinear and positive semidefinite. For $a\in\mathcal{P}_{k,\mathrm{sa}}$ we have $a^{*}=a$, so $\|a\|_{\lambda}$ is the seminorm $\|a\|_{\beta_{\lambda}}$ of Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences. Hence Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cauchy-schwarz, with $V=\mathcal{P}_{k,\mathrm{sa}}$ and $\beta=\beta_{\lambda}$, together with claim 8 of Properties of the Absolute Value in an Ordered Field (modulus and absolute value agree on $\mathbb{R}$), gives for $a,b\in\mathcal{P}_{k,\mathrm{sa}}$ and real $t$:

We also use three elementary facts.

(E1) If $s,t\in\mathbb{R}^{k}$ satisfy $0\le s_{l}\le t_{l}$ for all $l\in[k]$, then $\|s\|\le\|t\|$ for the Euclidean norm. Indeed, $s_{l}^{2}\le t_{l}^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $\sum_{l}s_{l}^{2}\le\sum_{l}t_{l}^{2}$ by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, that is, $\|s\|^{2}\le\|t\|^{2}$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. As both norms are nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\|s\|\le\|t\|$.

(E2) If $0\le\alpha\le\alpha'$ and $0\le\beta\le\beta'$ are real, then $\alpha\beta\le\alpha'\beta\le\alpha'\beta'$ by claim 5 of Elementary Arithmetic in an Ordered Field. Moreover, for real $t$ the absolute value $|t|$ is $t$ or $-t$ by claim 1 of Properties of the Absolute Value in an Ordered Field, so $|t|^{2}=t^{2}$.

(P) Let $\mu,\nu\in\Sigma_{m}$, say $\mu\in\Sigma_{m,R_{1}}$ and $\nu\in\Sigma_{m,R_{2}}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, and put $R=R_{1}+R_{2}$, so that $R>0$ by claim 3 of Elementary Order Arithmetic in an Ordered Field applied to $0<R_{1}$ and $0\le R_{2}$. Then $R_{1},R_{2}\le R$ by claim 3 of Elementary Arithmetic in an Ordered Field, so $\mu,\nu\in\Sigma_{m,R}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained there is an optimal $\gamma\in\Pi(\mu,\nu)$, that is, $I(\gamma)=W_{2}(\mu,\nu)^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. Moreover, by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance and Existence and Uniqueness of the Nonnegative Square Root, $W_{2}(\mu,\nu)^{2}$ is the infimum of the costs of couplings, so $W_{2}(\mu,\nu)^{2}\le I(\gamma')$ for every $\gamma'\in\Pi(\mu,\nu)$. This holds for laws of any number of variables.

For $k\in\mathbb{N}$ write $\mathrm{pr}^{1}_{k},\mathrm{pr}^{2}_{k},D_{k}$ for the coordinate data with $d=k$. By Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate, $\sigma_{\mathrm{pr}^{\epsilon}_{k}}=\iota^{\epsilon}$ ($\epsilon=1,2$), and by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §cost-identity, $\gamma(\Delta_{k})=M(\gamma\circ\sigma_{D_{k}})$ for every tracial state $\gamma$ on $\mathcal{P}_{2k}$.

Proof of clause 1 (Affine push-forwards). Let $T=(A,c)$ and $\mu,\nu\in\Sigma_{m}$. Then $\mu\circ\sigma_{T},\nu\circ\sigma_{T}\in\Sigma_{n}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint. Let $\gamma\in\Pi(\mu,\nu)$ be optimal, by (P). Each element of $[2n]$ is either $i$ or $n+i$ for exactly one $i\in[n]$, and likewise for $[2m]$. Define the affine datum $\tilde T=(\tilde A,\tilde c)$ from $2m$ to $2n$ variables by $\tilde A_{ij}=\tilde A_{n+i,m+j}=A_{ij}$, $\tilde A_{i,m+j}=\tilde A_{n+i,j}=0$ and $\tilde c_{i}=\tilde c_{n+i}=c_{i}$ ($i\in[n]$, $j\in[m]$). Let $T_{0}=(A,0)$, an affine datum from $m$ to $n$ variables with $\lVert T_{0}\rVert=\lVert T\rVert$.

Entries. Let $i\in[n]$ and $l\in[2m]$. For $\epsilon=1,2$ let $P^{\epsilon}_{n}$ denote the matrix of $\mathrm{pr}^{\epsilon}_{n}$ and $P^{\epsilon}_{m}$ the matrix of $\mathrm{pr}^{\epsilon}_{m}$. In each of the following sums at most one summand is nonzero, so claim 7 of Properties of Finite Sums gives

Similarly, $(AP^{1}_{m})_{il}=\sum_{j=1}^{m}A_{ij}(P^{1}_{m})_{jl}$ equals $A_{il}$ if $l\le m$ and $0$ if $l=m+j'$ with $j'\in[m]$. Likewise $(AP^{2}_{m})_{il}$ equals $0$ if $l\le m$ and $A_{ij'}$ if $l=m+j'$. Comparing with the definition of $\tilde A$ gives $P^{\epsilon}_{n}\tilde A=AP^{\epsilon}_{m}$ for $\epsilon=1,2$.

Composites. By Affine Data and Affine Substitutions of Noncommutative Polynomials §composite, and since $\sum_{j}A_{ij}\cdot0=0$ and $\sum_{k}(P^{\epsilon}_{n})_{ik}\cdot0=0$, we have $\mathrm{pr}^{\epsilon}_{n}\circ\tilde T=(P^{\epsilon}_{n}\tilde A,P^{\epsilon}_{n}\tilde c)$ and $T\circ\mathrm{pr}^{\epsilon}_{m}=(AP^{\epsilon}_{m},c)$. These are equal by the entries above. Likewise $D_{n}\circ\tilde T=((P^{1}_{n}-P^{2}_{n})\tilde A,(P^{1}_{n}-P^{2}_{n})\tilde c)$ and $T_{0}\circ D_{m}=(A(P^{1}_{m}-P^{2}_{m}),0)$. By claims 2 and 3 of Properties of Finite Sums, $((P^{1}_{n}-P^{2}_{n})\tilde A)_{il}=(P^{1}_{n}\tilde A)_{il}-(P^{2}_{n}\tilde A)_{il}$, $(A(P^{1}_{m}-P^{2}_{m}))_{il}=(AP^{1}_{m})_{il}-(AP^{2}_{m})_{il}$ and $((P^{1}_{n}-P^{2}_{n})\tilde c)_{i}=c_{i}-c_{i}=0$. Hence $D_{n}\circ\tilde T=T_{0}\circ D_{m}$.

A coupling of the push-forwards. Put $\tilde\gamma=\gamma\circ\sigma_{\tilde T}$, a tracial state on $\mathcal{P}_{2n}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint. By Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §composition, for $\epsilon=1,2$,

Since $\gamma\circ\iota^{1}=\mu$ and $\gamma\circ\iota^{2}=\nu$, we get $\tilde\gamma\in\Pi(\mu\circ\sigma_{T},\nu\circ\sigma_{T})$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling.

Its cost. Put $\rho=\gamma\circ\sigma_{D_{m}}$, a tracial state on $\mathcal{P}_{m}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint. By Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §cost-identity, Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §composition and $D_{n}\circ\tilde T=T_{0}\circ D_{m}$,

For $i\in[n]$ let $b_{i}=a^{T_{0}}_{i}=0\cdot1+\sum_{j=1}^{m}A_{ij}x_{j}$. It is self-adjoint by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, and $\sigma_{T_{0}}(x_{i}x_{i})=b_{i}b_{i}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism. Hence $I(\tilde\gamma)=\sum_{i=1}^{n}\|b_{i}\|_{\rho}^{2}$ and $I(\gamma)=M(\rho)=\sum_{j=1}^{m}\|x_{j}\|_{\rho}^{2}$.

The estimate. Fix $i\in[n]$. The partial sums $\sum_{j=1}^{k}A_{ij}x_{j}$ are self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. By induction on $k\in[m]$ along the recursion of claim 1 of both finite-sum lemmas, using (S) and the compatibility of the order with addition, $\|\sum_{j=1}^{k}A_{ij}x_{j}\|_{\rho}\le\sum_{j=1}^{k}|A_{ij}|\,\|x_{j}\|_{\rho}$. Now let $\alpha=(|A_{i1}|,\dots,|A_{im}|)$ and $q=(\|x_{1}\|_{\rho},\dots,\|x_{m}\|_{\rho})$ in $\mathbb{R}^{m}$. By Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, claim 3 of Properties of the Absolute Value in an Ordered Field and Cauchy-Schwarz Inequality for the Euclidean Dot Product,

Both ends are nonnegative. So claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and (E2) give

Summing over $i$ by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, and using claim 3 of Properties of Finite Sums and Affine Data and Affine Substitutions of Noncommutative Polynomials §norm, we get $I(\tilde\gamma)\le\lVert T\rVert^{2}M(\rho)=\lVert T\rVert^{2}I(\gamma)=\bigl(\lVert T\rVert W_{2}(\mu,\nu)\bigr)^{2}$.

Conclusion. By (P) for the laws $\mu\circ\sigma_{T},\nu\circ\sigma_{T}\in\Sigma_{n}$ and the coupling $\tilde\gamma$, $W_{2}(\mu\circ\sigma_{T},\nu\circ\sigma_{T})^{2}\le I(\tilde\gamma)\le(\lVert T\rVert W_{2}(\mu,\nu))^{2}$. Both $W_{2}(\mu\circ\sigma_{T},\nu\circ\sigma_{T})$ and $\lVert T\rVert W_{2}(\mu,\nu)$ are nonnegative, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives clause 1.

Proof of clause 2 (Moments). Let $\mu,\nu\in\Sigma_{m}$, put $W=W_{2}(\mu,\nu)$, let $\gamma\in\Pi(\mu,\nu)$ be optimal by (P), and write $\|\cdot\|=\|\cdot\|_{\gamma}$. For $i\in[m]$ put $y_{i}=x_{m+i}$ and $z_{i}=x_{i}-y_{i}$, self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism, $\iota^{1}(x_{i})=x_{i}$, $\iota^{2}(x_{i})=y_{i}$, $\iota^{1}(x_{i}x_{j})=x_{i}x_{j}$ and $\iota^{2}(x_{i}x_{j})=y_{i}y_{j}$. As $\gamma\circ\iota^{1}=\mu$ and $\gamma\circ\iota^{2}=\nu$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling), for $i,j\in[m]$ we get

By Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost and claim 4 of Properties of Finite Sums of Vectors, $W^{2}=I(\gamma)=\sum_{i=1}^{m}\gamma(z_{i}z_{i})=\sum_{i=1}^{m}\|z_{i}\|^{2}$. Claim 6 of Properties of Finite Sums and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field then give $\|z_{i}\|\le W$, $\|x_{i}\|\le M(\mu)^{1/2}$ and $\|y_{i}\|\le M(\nu)^{1/2}$ for all $i\in[m]$.

First moments. $\|1\|^{2}=\gamma(1^{*}1)=\gamma(1)=1$, so $\|1\|=1$. Here $1^{*}1=1$ because $1^{*}=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and $1\cdot1=1$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and $\gamma(1)=1$ by condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state. By linearity of $\gamma$ and (S) with $\lambda=\gamma$, $|\mathrm{m}_{i}(\mu)-\mathrm{m}_{i}(\nu)|=|\gamma(z_{i}1)|\le\|z_{i}\|\,\|1\|\le W$.

Second moment. Let $u,v,w\in\mathbb{R}^{m}$ have entries $u_{i}=\|x_{i}\|$, $v_{i}=\|y_{i}\|$ and $w_{i}=\|z_{i}\|$. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and the uniqueness in Existence and Uniqueness of the Nonnegative Square Root, $\|u\|=M(\mu)^{1/2}$, $\|v\|=M(\nu)^{1/2}$ and $\|w\|=W$. Since $x_{i}=y_{i}+z_{i}$ and $y_{i}=x_{i}+(-1)z_{i}$, (S) gives $u_{i}\le v_{i}+w_{i}$ and $v_{i}\le u_{i}+w_{i}$, and these are the entries of $v+w$ and $u+w$ by Sum of Points of $\mathbb{R}^n$. By (E1) and claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\|u\|\le\|v+w\|\le\|v\|+W$ and $\|v\|\le\|u\|+W$. Thus $-W\le\|u\|-\|v\|\le W$ by claim 3 of Elementary Arithmetic in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field. So $|M(\mu)^{1/2}-M(\nu)^{1/2}|\le W$ by claim 6 of Properties of the Absolute Value in an Ordered Field.

Quadratic moments. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $z_{i}x_{j}+y_{i}z_{j}=x_{i}x_{j}-y_{i}x_{j}+y_{i}x_{j}-y_{i}y_{j}=x_{i}x_{j}-y_{i}y_{j}$. So $\mathrm{m}_{ij}(\mu)-\mathrm{m}_{ij}(\nu)=\gamma(z_{i}x_{j})+\gamma(y_{i}z_{j})$, a sum of two real numbers by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §pairing. By claim 5 of Properties of the Absolute Value in an Ordered Field, (S) and (E2),

Proof of clause 3 (Cauchy sequences). By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric, $W_{2}$ is the metric of $\Sigma_{m}$ and of $\Sigma_{n}$, and Cauchy Sequence in a Metric Space applies. Let $\varepsilon>0$ be real.

Push-forwards. We have $\lambda_{k}\circ\sigma_{T}\in\Sigma_{n}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint. The number $\lVert T\rVert+1$ is positive by claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field. Choose $N$ with $W_{2}(\lambda_{k},\lambda_{l})<\varepsilon(\lVert T\rVert+1)^{-1}$ for $k,l\ge N$. Then clause 1, claim 5 of Elementary Arithmetic in an Ordered Field and claim 10 of Elementary Order Arithmetic in an Ordered Field give $W_{2}(\lambda_{k}\circ\sigma_{T},\lambda_{l}\circ\sigma_{T})\le\lVert T\rVert W_{2}(\lambda_{k},\lambda_{l})\le(\lVert T\rVert+1)W_{2}(\lambda_{k},\lambda_{l})<\varepsilon$.

First moments. Choose $N$ with $W_{2}(\lambda_{k},\lambda_{l})<\varepsilon$ for $k,l\ge N$. By clause 2, $|\mathrm{m}_{i}(\lambda_{k})-\mathrm{m}_{i}(\lambda_{l})|<\varepsilon$ for $k,l\ge N$, so $(\mathrm{m}_{i}(\lambda_{k}))_{k}$ is Cauchy in the sense of Cauchy Sequence of Real Numbers.

Quadratic moments. Choose $N_{0}$ with $W_{2}(\lambda_{k},\lambda_{l})<1$ for $k,l\ge N_{0}$, and put $B=M(\lambda_{N_{0}})^{1/2}+1$, so that $B\ge1>0$. For $k\ge N_{0}$, clause 2 and claim 3 of Properties of the Absolute Value in an Ordered Field give $M(\lambda_{k})^{1/2}\le M(\lambda_{N_{0}})^{1/2}+W_{2}(\lambda_{k},\lambda_{N_{0}})\le B$. Hence, for $k,l\ge N_{0}$, clause 2 and claim 5 of Elementary Arithmetic in an Ordered Field give $|\mathrm{m}_{ij}(\lambda_{k})-\mathrm{m}_{ij}(\lambda_{l})|\le2B\,W_{2}(\lambda_{k},\lambda_{l})$. By claims 5, 7 and 10 of Elementary Order Arithmetic in an Ordered Field, $\delta=\varepsilon(2B)^{-1}>0$. Choose $N_{1}$ with $W_{2}(\lambda_{k},\lambda_{l})<\delta$ for $k,l\ge N_{1}$, and let $N$ be the larger of $N_{0}$ and $N_{1}$. For $k,l\ge N$ we get $|\mathrm{m}_{ij}(\lambda_{k})-\mathrm{m}_{ij}(\lambda_{l})|\le2B\,W_{2}(\lambda_{k},\lambda_{l})<2B\delta=\varepsilon$, so $(\mathrm{m}_{ij}(\lambda_{k}))_{k}$ is Cauchy by Cauchy Sequence of Real Numbers.

Proof of clause 4 (Cost). By Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate, $\gamma\circ\iota^{\epsilon}=\gamma\circ\sigma_{\mathrm{pr}^{\epsilon}_{d}}$ for $\epsilon=1,2$. This lies in $\Sigma_{d}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, applied to the affine datum $\mathrm{pr}^{\epsilon}_{d}$ from $2d$ to $d$ variables and the law $\gamma\in\Sigma_{2d}$. Since $\gamma$ is a tracial state on $\mathcal{P}_{2d}$ whose marginals $\gamma\circ\iota^{1}$ and $\gamma\circ\iota^{2}$ are the given laws, $\gamma\in\Pi(\gamma\circ\iota^{1},\gamma\circ\iota^{2})$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling. Then (P), with $d$ in place of $m$, gives $W_{2}(\gamma\circ\iota^{1},\gamma\circ\iota^{2})^{2}\le I(\gamma)$.