Isoperimetric inequality for non-spherical multivariate Gaussian

Question

Disclaimer: Sorry in advance, if the question is not very reasonable. Recently (like a few days ago...), I've started studying isoperimetric inequalities, and my thoughts on the subject are rather cloudy with lots of black holes.

Setup

So, the classical Gaussian Isoperimetric inequality states that

[GIPI] if $d\gamma_d(x):= (2\pi)^{-d/2}\exp(-\|x\|_2^2/2)dx$ is the Gaussian measure on $\mathbb R^d$ and $B$ is a Borel subset of $\mathbb R^d$, then $$ \gamma_d(B_\epsilon) \ge \Phi(\Phi^{-1}(\gamma_d(B)) + \epsilon),\;\forall \epsilon > 0 $$

where:

• $B_\epsilon := \{x \in \mathbb R^d | \|x-A\|_2^2 \le \epsilon\}$ is $\epsilon$-neighborhood/blowup of $B$
• the function $\Phi(z): = \int_{-\infty}^zd\gamma_1(z)$ defines the CDF of the 1D standard Gaussian and $\Phi^{-1}$ is its inverse.

A corollary to this inequality (see Otto et Villani 2000, for example) says that

[GIPI Corollary] if $\epsilon \ge \sqrt{2\log(1/\gamma_d(B))}$, then $$ \gamma_d(B_\epsilon) \ge 1 - \exp(-\frac{1}{2}(\epsilon-\sqrt{2\log(1/\gamma_d(B))})^2). $$

I'm interested in "simple" generalizations of [GIPI] and [GIPI Corollary].

Question 1

I know there are works (e.g Otto et Villani 2000, Bobkov 1999, etc.) which generalize this to more general log-concave distributions $d\mu = e^{-V}dx$ (coupled with even more general curved manifolds with lower bounded Ricci curvature), but it's hard to get one's hand on precise formulae (e.g involving the modulus of strong-convexity of the potential $V$, etc.). As a concrete example, suppose $V$ is twice continuously differentiable with $\operatorname{Hess}(V) \succeq C \operatorname{I}_d$, what analogues of [GIPI] and [GIPI Corollary] do we get ?

Question 2

In particular, what analogues of [GIPI] and [GIPI Corollary] do we get when we instead consider an non-spherical Gaussian distribution $d\gamma_{\mu,\Sigma} := (2\pi\operatorname{det}(\Sigma))^{-d/2}\exp(-\frac{1}{2}x^T\Sigma^{-1}x)$, with mean $\mu$ and covariance matrix $\Sigma$ (positive definite) ?

Observation

It turns out that explicit answers to both questions can be derived from Corollary 3.2 Bobkov et al. For example, we have a multi-variate Gaussian version of [GIPI Corollary] by simply replacing the 2 appearing in the square-root and the fraction (i.e $\frac{1}{2}$) with $2$ $\times$ the smallest eigenvalue of covariance matrix $\Sigma$. I can writeup a self-contained answer, but I don't know if this is a good idea since I asked the question in the first place.

Answer 1

So, the works of Bobkov et al. (1999), Otto et Villani (2000), and even more recently Otto et Reznikoff (2006), contain elements of solution to the questions I asked. I'll provide a quasi self-contained answer, hoping it will be helpful to a general public (like myself) without kungfu-panda-level knowledge on concentration inequalities.

N.B.: By "Riemannian manifold" I'll actually mean a smooth Riemannian manifold which is complete with the geodesic distance

$$d_p(x,y) := \inf_{\omega \in \mathcal C^1([0, 1], M) | \omega(0)=x,\omega(1) = y}\sqrt{\int_0^1\|\dot\omega(t)\|^2dt}. $$

Definition (LSI). A probability distribution $\mu$ is said to satisfy a Logarithmic Sobolev Inequality with constant $\rho > 0$, written $\operatorname{LSI}(\rho)$, if for every probability distribution $\nu$ absolutely continuous w.r.t $\mu$, it holds that $$ H(\mu\|\nu) \le \frac{1}{2\rho}F(\mu\|\nu), $$ where $H(\mu\|\nu)$ is relative entropy and $F(\mu\|\nu)$ is relative Fisher information.

Definition (LC). A probability distribution $\mu$ on a Riemannian nanifold $M$ is called log-concave with constant $\rho$, written $\operatorname{LC}(\rho)$, if there exists a a twice continuous differentiable potential field $V$ on $M$ such that $\mu = e^{-V(x)}dx$ and $$ \operatorname{Hess}(V) + \operatorname{Ricc}(M) \succeq \rho I_d $$ Definition (GIPIC). A probability distribution $\mu$ is said to satisfy GIPIC (Gaussian IsoPerimetic Inequality Corollary) with constant $\rho > 0$, if for every Borel set $B \subseteq M$ it holds that $$ \mu(B_\epsilon) \ge 1 - \exp\left(-\frac{\rho}{2}(\epsilon-\sqrt{(2/\rho)\log(1/\mu(B))})^2\right),\;\forall \epsilon \ge \sqrt{(2/\rho)\log(1/\mu(B))}, $$ where $B_\epsilon$ is the $\epsilon$-neighborhood of $B$.

And now, the real deal

Theorem [Corollary 3.2 of Bobkov et al. 1999, Corrolary 1.1 of Otto et Villani 2000]. On any Riemannian manifold, it holds that $$\operatorname{LC}(\rho) \implies \operatorname{LSI}(\rho) \implies \operatorname{GIPIC}(\rho). $$

Application

If $M = \mathbb R^d$ and $\mu = d\gamma_{c,\Sigma} \propto e^{-V(x)}dx$ where $V(x) = \frac{1}{2}(x-c)^T\Sigma^{-1}(x-c)$, for some p.s.d matrix $\Sigma$ with largest singular value $\sigma^2$ and vector $c \in \mathbb R^d$, then $\operatorname{Hess}(V) + \operatorname{Ricc}(M) \succeq (1/\sigma^2 + 0)I_d = (1/\sigma^2)I_d$. Thus each multivariate Gaussian $d\gamma_{c,\Sigma}$ distribution on $\mathbb R^d$ satisfies GIPIC($1/\sigma^2$), more precisely, it holds that $$ \gamma_{c,\Sigma}(B_\epsilon) \ge 1 - \exp\left(-\frac{1}{2\sigma^2}(\epsilon-\sigma\sqrt{2\log(1/\gamma_{c,\Sigma}(B))})^2\right),\;\forall \epsilon \ge \sigma\sqrt{2\log(1/\gamma_{c,\Sigma}(B))}. $$