Proof of Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score

Each result cited is universally quantified over the data in its own statement.

Throughout, $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$, $\lambda$ and $\mu_{*}$ are as in the statement. By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair we have $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$, so $\mu_{*}\in\mathcal{D}$ and $\mathcal{E}(\mu_{*})$ is a real number.

Step 1 (Optimal couplings and their cost). Let $\alpha,\beta\in\mathcal{P}^{a}_{\rho}$. By The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected the ordered pair $(\alpha,\beta)$ is noise-connected, so $W_{a}(\alpha,\beta)$ is defined and is a nonnegative real number by The Noise Wasserstein Distance §distance; by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §optimal there is a noise-optimal coupling $\pi\in\Pi^{a}(\alpha,\beta)$, and by Noise-Optimal Couplings §optimal it satisfies $I^{a}(\pi)=W_{a}(\alpha,\beta)^{2}$. Since $W_{a}(\alpha,\beta)$ is nonnegative and its square is $I^{a}(\pi)$, the uniqueness of nonnegative square roots in Existence and Uniqueness of the Nonnegative Square Root gives $\sqrt{I^{a}(\pi)}=W_{a}(\alpha,\beta)$.

Step 2 (Talagrand inequality). Let $\nu\in\mathcal{D}$. Both $\mu_{*}$ and $\nu$ lie in $\mathcal{P}^{a}_{\rho}$, so by Step 1 (with $\alpha=\mu_{*}$, $\beta=\nu$) there is a noise-optimal coupling $\pi\in\Pi^{a}(\mu_{*},\nu)$ with $I^{a}(\pi)=W_{a}(\mu_{*},\nu)^{2}$. Since $\Sigma(\mu_{*})$ is the zero vector of $L^{2}(\mu_{*};X^{a})$, its norm $\lVert\Sigma(\mu_{*})\rVert_{\mu_{*}}$ is $0$, and the bound of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound, applied with $\mu_{*}$ in place of $\mu$, the coupling $\pi$ and $\eta=\Sigma(\mu_{*})$, gives $|\mathcal{J}^{a}(\Sigma(\mu_{*}),\pi)|\le0\cdot\sqrt{I^{a}(\pi)}=0$; hence $\mathcal{J}^{a}(\Sigma(\mu_{*}),\pi)=0$. Now $\lambda$-displacement convexity, Lambda-Displacement Convexity of a Noise Penalty Pair §convex, applied with $\mu_{*}\in\mathcal{D}_{\Sigma}$ in place of $\mu$, with $\nu\in\mathcal{D}$ and with the noise-optimal coupling $\pi$, gives

$$\mathcal{E}(\mu_{*})+0+\frac{\lambda}{2}\,W_{a}(\mu_{*},\nu)^{2}\le\mathcal{E}(\nu),$$

after substituting $I^{a}(\pi)=W_{a}(\mu_{*},\nu)^{2}$ and dropping the zero pairing, which is claim Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §talagrand. Since $\lambda$ is positive and $W_{a}(\mu_{*},\nu)^{2}\ge0$, the term $\frac{\lambda}{2}W_{a}(\mu_{*},\nu)^{2}$ is nonnegative, and therefore $\mathcal{E}(\mu_{*})\le\mathcal{E}(\nu)$.

Step 3 (HWI inequality). Let $\mu\in\mathcal{D}_{\Sigma}$, and write $w=W_{a}(\mu,\mu_{*})$ and $A=\lVert\Sigma(\mu)\rVert_{\mu}$, both nonnegative real numbers. Both $\mu$ and $\mu_{*}$ lie in $\mathcal{P}^{a}_{\rho}$, so by Step 1 (with $\alpha=\mu$, $\beta=\mu_{*}$) there is a noise-optimal coupling $\pi\in\Pi^{a}(\mu,\mu_{*})$ with $I^{a}(\pi)=w^{2}$ and $\sqrt{I^{a}(\pi)}=w$. By Lambda-Displacement Convexity of a Noise Penalty Pair §convex, applied with this $\mu\in\mathcal{D}_{\Sigma}$, with $\mu_{*}\in\mathcal{D}$ in place of $\nu$, and with $\pi$,

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{\lambda}{2}\,w^{2}\le\mathcal{E}(\mu_{*}).$$

By Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §bound, applied with $\eta=\Sigma(\mu)\in L^{2}(\mu;X^{a})$ and the coupling $\pi$, we have $-\mathcal{J}^{a}(\Sigma(\mu),\pi)\le|\mathcal{J}^{a}(\Sigma(\mu),\pi)|\le A\sqrt{I^{a}(\pi)}=A\,w$. Rearranging the displayed inequality,

$$\mathcal{E}(\mu)-\mathcal{E}(\mu_{*})\le-\mathcal{J}^{a}(\Sigma(\mu),\pi)-\frac{\lambda}{2}\,w^{2}\le A\,w-\frac{\lambda}{2}\,w^{2},$$

which is claim Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §hwi.

Step 4 (Log-Sobolev inequality). Let $\mu\in\mathcal{D}_{\Sigma}$, with $A$ and $w$ as in Step 3. Since $\lambda$ is positive, completing the square gives

$$A\,w-\frac{\lambda}{2}\,w^{2}=\frac{A^{2}}{2\lambda}-\frac{\lambda}{2}\Bigl(w-\frac{A}{\lambda}\Bigr)^{2}\le\frac{A^{2}}{2\lambda},$$

the subtracted term being nonnegative as a positive multiple of a square. Combined with Step 3 this gives $\mathcal{E}(\mu)-\mathcal{E}(\mu_{*})\le\frac{1}{2\lambda}\lVert\Sigma(\mu)\rVert_{\mu}^{2}$, which is claim Talagrand, HWI and Log-Sobolev Inequalities for a Uniformly Displacement Convex Noise Penalty Pair with a Point of Zero Score §log-sobolev.