{"id":17237,"date":"2023-04-06T14:13:27","date_gmt":"2023-04-06T12:13:27","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=17237"},"modified":"2024-01-14T16:45:50","modified_gmt":"2024-01-14T15:45:50","slug":"poincare-log-concave-laws","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2023\/04\/06\/poincare-log-concave-laws\/","title":{"rendered":"Poincar\u00e9 for log-concave laws"},"content":{"rendered":"
\"Andr\u00e9<\/a>
Andr\u00e9 Lichnerowicz (1915 - 1998). Differential Geometry, Global Analysis on Manifolds, and Mathematical Physics.<\/figcaption><\/figure>\n

This post is centered around a recent nice observation by Bo\u00e1z Klartag on the Poincar\u00e9 constant of uniformly log-concave measures, at the heart of his recent investigation of the Kannan-Lov\u00e1sz-Simonovits (KLS) conjecture, that we do not discuss here.<\/p>\n

A famous observation due to Andr\u00e9 Lichnerowicz in differential geometry states that if a compact connected Riemannian manifold of dimension \\( {n\\geq2} \\) has Ricci curvature uniformly bounded below by some real number \\( {\\rho>0} \\), then the spectral gap \\( {\\lambda} \\), or first non-zero eigenvalue of minus the Laplace operator on the manifold, satisfies<\/p>\n

\\[ \\lambda\\geq\\frac{n}{n-1}\\rho. \\]<\/p>\n

This simply follows by testing the Bochner commutation-curvature formula on an eigenfunction associated to the spectral gap, a method still at the heart of the Klartag observation. Equality is achieved for spheres, and for the unit sphere we have \\( {\\lambda=n} \\) while we can take \\( {\\rho=n-1} \\). The Lichnerowicz inequality is essentially a comparison to spheres.<\/p>\n

This statement has an analogue for uniformly log-concave probability measures on \\( {\\mathbb{R}^n} \\) : if \\( {\\mathrm{d}\\mu(x)=\\mathrm{e}^{-V(x)}\\mathrm{d}x} \\) on \\( {\\mathbb{R}^n} \\) with, for some \\( {\\rho>0} \\), \\( {\\nabla^2V(x)\\geq\\rho\\mathrm{Id}} \\) as quadratic forms for all \\( {x\\in\\mathbb{R}^n} \\), then the spectral gap \\( {\\lambda} \\) of the Markov diffusion operator \\( {\\Delta-\\langle\\nabla V,\\nabla\\rangle} \\) satisfies<\/p>\n

\\[ \\lambda\\geq\\rho. \\]<\/p>\n

This inequality is obtained below by following the Lichnerowicz argument, via the Bochner commutation-curvature formula on an eigenfunction. Equality is achieved for the Gaussian case \\( {V(x)=\\frac{\\rho}{2}\\|x\\|^2} \\), and the inequality is this time essentially a comparison to Gaussians.<\/p>\n

This inequality can also be deduced from the Bakry-\u00c9mery curvature-dimension criterion, which is an abstraction of the Bochner commutation-curvature formula approach. For the pleasure, we also present, at the end of the post, a derivation of the Brascamp-Lieb inequality from the Bochner commutation-curvature formula, via a Helffer-Sj\u00f6strand representation of the covariance, which is of independent interest.<\/p>\n

This post is mostly devoted to the following improvement by Bo\u00e1z Klartag :<\/p>\n

\\[ \\lambda\\geq\\sqrt{\\frac{\\rho}{\\|\\Sigma\\|_{\\mathrm{op}}}}\\geq\\rho, \\]<\/p>\n

where \\( {\\|\\Sigma\\|_{\\mathrm{op}}} \\) is the operator norm of the covariance matrix of \\( {\\mu} \\). On the opposite side, note that we always have \\( {\\lambda\\leq1\/\\|\\Sigma\\|_{\\mathrm{op}}} \\) regardless of the log-concavity of \\( {\\mu} \\), see below. Equality is achieved in the above inequalities in the Gaussian case for which \\( {\\Sigma=\\frac{1}{\\rho}\\mathrm{Id}} \\) and \\( {\\lambda=\\rho} \\).<\/p>\n

The first Klartag inequality reads<\/p>\n

\\[ \\lambda^2\\|\\Sigma\\|_{\\mathrm{op}}\\geq\\rho, \\]<\/p>\n

and reminds the uncertainty principle in harmonic analysis : for a given lower bound on the curvature, we cannot have in the same time a small spectral gap and a small operator norm for the covariance. The second Klartag inequality reads<\/p>\n

\\[ \\rho\\|\\Sigma\\|_{\\mathrm{op}}\\leq1, \\]<\/p>\n

and can be interpreted as follows : we cannot have in the same time a high value for the lower bound on the curvature and for the operator norm of the covariance.<\/p>\n

Poincar\u00e9 inequality and Poincar\u00e9 constant.<\/b> The Poincar\u00e9 constant \\( {c_{\\mathrm{P}}(\\mu)} \\) of a probability measure \\( {\\mu} \\) on \\( {\\mathbb{R}^n} \\) is the smallest constant \\( {c} \\) such that for all \\( {f\\in\\mathcal{C}^\\infty_c} \\),<\/p>\n

\\[ \\mathrm{Var}_\\mu(f) :=\\int\\Bigr(f-\\int f\\mathrm{d}\\mu\\Bigr)^2\\mathrm{d}\\mu \\leq c\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu. \\]<\/p>\n

An approximation argument based on cutoff and smoothing allows typically to extend the inequality to \\( {f\\in\\mathrm{H}^1(\\mu)} \\) where \\( {\\mathrm{H}^1(\\mu)=\\mathrm{W}^{1,2}(\\mu)} \\) is the Sobolev space of square integrable functions with square integrable weak derivative. Both the left and right hand sides of the inequality vanish if and only if \\( {f} \\) is constant \\( {\\mu} \\) almost surely, and<\/p>\n

\\[ \\frac{1}{c_{\\mathrm{P}}(\\mu)} =\\inf \\Bigr\\{\\frac{\\displaystyle\\int|\\nabla f|^2\\mathrm{d}\\mu}{\\mathrm{Var}_\\mu(f)}: f\\text{ not constant}\\Bigr\\}. \\]<\/p>\n

Let us introduce the mean vector<\/b> or barycenter<\/b> \\( {m={(m_i)}_{1\\leq i\\leq n}} \\) of \\( {\\mu} \\) defined by<\/p>\n

\\[ m:=\\int x\\mathrm{d}\\mu(x) \\]<\/p>\n

and the covariance matrix<\/b> \\( {{(\\Sigma_{i,j})}_{1\\leq i,j\\leq n}} \\) of \\( {\\mu} \\) defined by<\/p>\n

\\[ \\Sigma_{ij} = \\int x_ix_j\\mathrm{d}\\mu(x) -\\Bigr(\\int x_i\\mathrm{d}\\mu(x)\\Bigr) \\Bigr(\\int x_j\\mathrm{d}\\mu(x)\\Bigr) =\\int(x_i-m_i)(x_j-m_j)\\mathrm{d}\\mu(x). \\]<\/p>\n

For all \\( {u\\in\\mathbb{R}^n} \\) with \\( {\\|u\\|=1} \\), the linear function<\/b> \\( {g_u(x):=\\langle x,u\\rangle=\\sum_{i=1}^n u_ix_i} \\) satisfies<\/p>\n

\\[ \\int g_u\\mathrm{d}\\mu=\\langle m,u\\rangle \\quad\\text{and}\\quad \\int g_u^2\\mathrm{d}\\mu =\\langle\\Sigma u,u\\rangle+\\langle b,u\\rangle^2 \\quad\\text{and}\\quad \\|\\nabla g_u(x)\\|^2=\\|u\\|^2=1, \\]<\/p>\n

hence by Poincar\u00e9 inequality,<\/p>\n

\\[ \\langle\\Sigma u,u\\rangle =\\mathrm{Var}_\\mu(g_u) \\leq c_{\\mathrm{P}}, \\]<\/p>\n

Introducing the operator norm<\/b> \\( { \\|\\Sigma\\|_{\\mathrm{op}}:=\\sup_{\\|u\\|=1}\\|\\Sigma u\\|=\\sup_{\\|u\\|=1}\\langle\\Sigma u,u\\rangle} \\), this gives<\/p>\n

\\[ \\|\\Sigma\\|_{\\mathrm{op}}\\leq c_{\\mathrm{P}}. \\]<\/p>\n

Equality is achieved when \\( {\\mu} \\) is Gaussian, as we can check by scaling with \\( {\\sqrt{\\Sigma}} \\), tensorization, and expansion of \\( {L^2(\\mathcal{N}(0,1))} \\) in terms of Hermite orthogonal polynomials.<\/p>\n

The Kannan-Lov\u00e1sz-Simonovits (KLS) conjecture states that the opposite bound<\/p>\n

\\[ c_{\\mathrm{P}}\\leq c\\|\\Sigma\\|_{\\mathrm{op}} \\]<\/p>\n

holds up to a universal constant \\( {c} \\) provided that \\( {\\mu} \\) is log-concave (meaning that \\( {V} \\) is convex), in other words the Poincar\u00e9 constant of log-concave measures can be checked on linear test functions. The best bound at the time of writing, due to Klartag, and based on his discovery of an improved Lichnerowicz bound, is<\/p>\n

\\[ c_{\\mathrm{P}}\\leq c\\sqrt{\\log(n)}\\|\\Sigma\\|_{\\mathrm{op}}. \\]<\/p>\n

Spectral gap of Markov diffusion operator.<\/b> Suppose from now on that \\( {\\mu} \\) writes<\/p>\n

\\[ \\mathrm{d}\\mu(x)=\\mathrm{e}^{-V(x)}\\mathrm{d}x \\]<\/p>\n

with \\( {V\\in\\mathcal{C}^2(\\mathbb{R}^n\\rightarrow\\mathbb{R})} \\). The associated Markov diffusion operator<\/b> is<\/p>\n

\\[ \\mathrm{L}f:=\\Delta f-\\langle\\nabla V,\\nabla f\\rangle. \\]<\/p>\n

We have the integration by parts<\/b> formula, for all \\( {f,g\\in\\mathcal{C}^\\infty_c} \\),<\/p>\n

\\[ \\int f(-\\mathrm{L})g\\mathrm{d}\\mu =\\int g(-\\mathrm{L})f\\mathrm{d}\\mu =\\int\\langle \\nabla f,\\nabla g\\rangle\\mathrm{d}\\mu, \\]<\/p>\n

in particular, \\( {f=g} \\) and \\( {g\\rightarrow1} \\) give<\/p>\n

\\[ \\int f(-\\mathrm{L})f\\mathrm{d}\\mu =\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu \\quad\\text{and}\\quad \\int \\mathrm{L}f\\mathrm{d}\\mu =0. \\]<\/p>\n

In \\( {L^2(\\mu)} \\) the unbounded operator \\( {-\\mathrm{L}} \\) is non-negative since \\( {\\mathrm{L}} \\) is Markov, and its kernel contains only the constant functions. We have<\/p>\n

\\[ \\frac{1}{c_{\\mathrm{P}}} = \\inf\\Bigr\\{\\frac{\\displaystyle\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu}{\\mathrm{Var}_\\mu(f)}:f\\text{ not constant}\\Bigr\\} = \\inf_{\\substack{\\int f\\mathrm{d}\\mu=0\\\\\\int f^2\\mathrm{d}\\mu=1}}\\int f(-\\mathrm{L})f\\mathrm{d}\\mu, \\]<\/p>\n

Moreover \\( {-\\mathrm{L}} \\) has discrete spectrum formed by eigenvalues \\( {0=\\lambda_0<\\lambda_1<\\cdots} \\) and \\( {\\lambda:=\\lambda_1-\\lambda_0=\\lambda_1} \\) is the spectral gap<\/b> of \\( {-\\mathrm{L}} \\). If \\( {f} \\) is an eigenfunction<\/b> of \\( {-\\mathrm{L}} \\) with eigenvalue<\/b> \\( {\\lambda_1} \\), we can assume by scaling that \\( {\\int f^2\\mathrm{d}\\mu=1} \\), while \\( {-\\mathrm{L}f=\\lambda f} \\) gives \\( {\\int f\\mathrm{d}\\mu=0} \\), thus<\/p>\n

\\[ \\frac{1}{c_{\\mathrm{P}}} =\\lambda. \\]<\/p>\n

Bochner formula.<\/b> It is the commutator-curvature formula<\/p>\n

\\[ \\mathrm{L}\\nabla-\\nabla \\mathrm{L}=\\nabla^2V\\nabla. \\]<\/p>\n

By taking the inner product with \\( {\\nabla} \\) and using the integration by parts, we get<\/p>\n

\\[ \\begin{array}{rcl} \\displaystyle\\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu &=&\\displaystyle\\int\\langle \\mathrm{L}\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu -\\int\\langle\\nabla f,\\nabla \\mathrm{L}f\\rangle\\mathrm{d}\\mu\\\\ &=&\\displaystyle-\\int\\|\\nabla^2f\\|_{\\mathrm{HS}}^2\\mathrm{d}\\mu +\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu, \\end{array} \\]<\/p>\n

where \\( {\\|\\nabla^2f\\|_{\\mathrm{HS}}^2=\\mathrm{Tr}((\\nabla^2f)^2)=\\sum_{ij=1}^n(\\partial^2_{ij}f)^2} \\) is the squared Hilbert-Schmidt norm of the Hessian matrix of \\( {f} \\). In other words, we have obtained the mean Bochner formula<\/b><\/p>\n

\\[ \\int(\\mathrm{L}f)^2\\mathrm{d}\\mu = \\int\\|\\nabla^2f\\|_{\\mathrm{HS}}^2\\mathrm{d}\\mu +\\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu. \\]<\/p>\n

In Bakry-\u00c9mery theory, this is also the integrated \\( {\\Gamma_2} \\) formula \\( {\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu=\\int\\Gamma_2 f\\mathrm{d}\\mu} \\).<\/p>\n

If \\( {\\mu} \\) is uniformly log-concave<\/b> : for a constant \\( {\\rho>0} \\) and a convex \\( {C:\\mathbb{R}^n\\rightarrow\\mathbb{R}} \\),<\/p>\n

\\[ V(x)=\\frac{1}{2\\rho}\\|x\\|^2+C(x), \\]<\/p>\n

then \\( {\\nabla^2V\\geq\\rho \\mathrm{Id}} \\) as quadratic forms, and if \\( {f} \\) is an eigenfunction<\/b> of \\( {-\\mathrm{L}} \\) associated to the eigenvalue<\/b> \\( {\\lambda} \\) carrying the spectral gap of \\( {-\\mathrm{L}} \\) then we can assume by scaling that \\( {\\int f^2\\mathrm{d}\\mu=1} \\), and we get then from the mean Bochner formula<\/b> and integration by parts<\/b><\/p>\n

\\[ \\lambda^2 =\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu \\geq \\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu \\geq\\rho\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu =\\rho\\int f(-\\mathrm{L})f\\mathrm{d}\\mu =\\lambda\\rho, \\]<\/p>\n

hence the inequality<\/p>\n

\\[ \\frac{1}{c_{\\mathrm{P}}}=\\lambda\\geq\\rho, \\]<\/p>\n

which is a log-concave analogue of the Lichnerowicz inequality. This method of proof is essentially the one of Lichnerowicz. Equality is achieved in the Gaussian case \\( {C\\equiv0} \\).<\/p>\n

Klartag theorem.<\/b> It reinforces the previous result by incorporating an information on the covariance matrix. If \\( {\\mu} \\) is uniformly log-concave of constant \\( {\\rho>0} \\) then<\/p>\n

\\[ c_{\\mathrm{P}}\\leq\\sqrt{\\frac{\\|\\Sigma\\|_{\\mathrm{op}}}{\\rho}} \\leq\\frac{1}{\\rho}. \\]<\/p>\n

Proof of Klartag theorem.<\/b> Let \\( {f} \\) be an eigenfunction of \\( {-\\mathrm{L}} \\) associated to the eigenvalue \\( {\\lambda=1\/c_{\\mathrm{P}}} \\), namely \\( {-\\mathrm{L}f=\\lambda f=\\frac{1}{c_{\\mathrm{P}}}f} \\). We can assume by scaling that \\( {\\int f^2\\mathrm{d}\\mu=1} \\). Moreover \\( {-\\mathrm{L}f=\\lambda f} \\) gives \\( {\\int f\\mathrm{d}\\mu=-\\lambda^{-1}\\int \\mathrm{L}f\\mathrm{d}\\mu=0} \\). The Klartag lemma below gives<\/p>\n

\\[ \\int\\langle\\nabla V^2\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu \\leq\\lambda^3\\|\\Sigma\\|_{\\mathrm{op}}. \\]<\/p>\n

But since \\( {\\nabla^2V\\geq\\rho \\mathrm{Id}} \\) in the sense of quadratic forms, we get<\/p>\n

\\[ \\int\\langle\\nabla V^2\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu \\geq \\rho\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu =\\rho\\int f(-\\mathrm{L})f\\mathrm{d}\\mu =\\rho\\lambda, \\]<\/p>\n

hence the first inequality of the theorem<\/p>\n

\\[ \\lambda^2\\|\\Sigma\\|_{\\mathrm{op}}\\geq\\rho. \\]<\/p>\n

For the second inequality of the theorem, we use the fact that \\( {\\|\\Sigma\\|_{\\mathrm{op}}\\leq c_{\\mathrm{P}}=\\frac{1}{\\lambda}\\leq\\frac{1}{\\rho}} \\).<\/p>\n

Klartag lemma.<\/b> If \\( {f} \\) is an eigenfunction of \\( {-\\mathrm{L}} \\) associated to the eigenvalue \\( {\\lambda=1\/c_{\\mathrm{P}}} \\) in other words \\( {-\\mathrm{L}f=\\lambda f=(1\/c_{\\mathrm{P}})f} \\) and such that \\( {\\int f^2\\mathrm{d}\\mu=1} \\), then<\/p>\n

\\[ \\frac{1}{\\lambda} \\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu \\leq \\Bigr\\|\\int\\nabla f\\mathrm{d}\\mu\\Bigr\\|^2 =\\lambda^2\\Bigr\\|\\int xf(x)\\mathrm{d}\\mu(x)\\Bigr\\|^2 \\leq\\lambda^2\\|\\Sigma\\|_{\\mathrm{op}}. \\]<\/p>\n

Proof of Klartag lemma.<\/b> Let \\( {f} \\) be such that \\( {-\\mathrm{L}f=\\lambda f} \\) with \\( {\\lambda=1\/c_{\\mathrm{P}}} \\), and \\( {\\int f^2\\mathrm{d}\\mu=1} \\). By using the integration by parts, the mean Bochner formula, and the Poincar\u00e9 inequality for each \\( {\\partial_if} \\), \\( {1\\leq i\\leq n} \\), we get<\/p>\n

\\[ \\begin{array}{rcl} \\lambda^2 &=&\\displaystyle\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu =\\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu +\\int\\|\\nabla^2f\\|_{\\mathrm{HS}}^2\\mathrm{d}\\mu\\\\ &\\geq&\\displaystyle\\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu +\\lambda\\Bigr(\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu-\\Bigr\\|\\int\\nabla f\\mathrm{d}\\mu\\Bigr\\|^2\\Bigr)\\\\ &\\geq&\\displaystyle\\int\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu +\\lambda^2-\\lambda\\Bigr\\|\\int\\nabla f\\mathrm{d}\\mu\\Bigr\\|^2. \\end{array} \\]<\/p>\n

On the other hand, for any \\( {u\\in\\mathbb{R}^n} \\) such that \\( {\\|u\\|=1} \\), with \\( {g_u(x):=\\langle x,u\\rangle} \\),<\/p>\n

\\[ \\int\\langle\\nabla f,u\\rangle\\mathrm{d}\\mu =\\int\\langle\\nabla f,\\nabla g_u\\rangle\\mathrm{d}\\mu =-\\int (\\mathrm{L}f)g_u\\mathrm{d}\\mu =\\lambda\\int f(x)\\langle x,u\\rangle\\mathrm{d}\\mu(x). \\]<\/p>\n

But from this identity, using \\( {\\int f^2\\mathrm{d}\\mu=1} \\), \\( {\\int f\\mathrm{d}\\mu=0} \\), and the Cauchy-Schwarz inequality,<\/p>\n

\\[ \\begin{array}{rcl} \\displaystyle\\Bigr|\\int\\langle\\nabla f,u\\rangle\\mathrm{d}\\mu\\Bigr|^2 &=&\\displaystyle\\lambda^2\\Bigr|\\int f(x)(\\langle x,u\\rangle-\\langle m,u\\rangle)\\mathrm{d}\\mu(x)\\Bigr|^2\\\\ &\\leq&\\displaystyle\\lambda^2\\int(\\langle x,u\\rangle-\\langle m,u\\rangle)^2\\mathrm{d}\\mu(x)\\\\ &=&\\displaystyle\\lambda^2\\langle\\Sigma u,u\\rangle. \\end{array} \\]<\/p>\n

Helffer-Sj\u00f6strand, Brascamp-Lieb, Poincar\u00e9.<\/b> The covariance<\/b> of \\( {f} \\) and \\( {g} \\) is<\/p>\n

\\[ \\mathrm{Cov}_\\mu(f,g) :=\\int\\Bigr(f-\\int f\\mathrm{d}\\mu\\Bigr) \\Bigr(g-\\int g\\mathrm{d}\\mu\\Bigr)\\mathrm{d}\\mu. \\]<\/p>\n

For a fixed \\( {f} \\), let us seek for \\( {h} \\) depending on \\( {f} \\) such that for all \\( {g} \\),<\/p>\n

\\[ \\mathrm{Cov}_\\mu(f,g) =\\int\\langle\\nabla h,\\nabla g\\rangle\\mathrm{d}\\mu =-\\int (\\mathrm{L}h)g\\mathrm{d}\\mu. \\]<\/p>\n

Since \\( {\\int \\mathrm{L}h\\mathrm{d}\\mu=0} \\), we get<\/p>\n

\\[ \\int\\Bigr(f-\\int f\\mathrm{d}\\mu+\\mathrm{L}h\\Bigr)\\Bigr(g-\\int g\\mathrm{d}\\mu\\Bigr)\\mathrm{d}\\mu=0, \\]<\/p>\n

in other words \\( {f-\\int f\\mathrm{d}\\mu+\\mathrm{L}h} \\) is orthogonal to centered functions, and is thus constant, also the following Poisson equation<\/b> holds true:<\/p>\n

\\[ f-\\int f\\mathrm{d}\\mu=-\\mathrm{L}h. \\]<\/p>\n

Let us try to express \\( {\\nabla h} \\) in terms of \\( {\\nabla f} \\). By the Bochner formula<\/p>\n

\\[ \\nabla f=-\\nabla \\mathrm{L}h=-\\mathrm{L}\\nabla h \\]<\/p>\n

where \\( {\\mathrm{L}\\nabla h} \\) is the action of \\( {\\mathrm{L}} \\) on \\( {\\nabla h} \\), coordinate by coordinate, in other words the operator \\( {\\mathrm{L}} \\) acting on differential forms<\/b> just like the Laplacian in de Rham cohomology. Also \\( {h} \\) must be such that \\( {\\nabla h=(-\\mathrm{L})^{-1}\\nabla f} \\), which gives the Helffer-Sj\u00f6strand formula<\/b><\/p>\n

\\[ \\mathrm{Cov}_\\mu(f,g) =\\int\\langle(-\\mathrm{L})^{-1}(\\nabla f),\\nabla g\\rangle\\mathrm{d}\\mu. \\]<\/p>\n

Suppose from now on that \\( {\\mu} \\) is strictly log-concave in the sense that \\( {\\nabla^2V>0} \\) everywhere in the sense of quadratic forms. By the Bochner formula, as functional quadratic forms on differential forms<\/b> we have<\/p>\n

\\[ (-\\mathrm{L})\\geq\\nabla^2V, \\quad\\text{hence}\\quad (-\\mathrm{L})^{-1}\\leq(\\nabla^2V)^{-1}. \\]<\/p>\n

Combined with the Helffer-Sj\u00f6strand representation of the variance, we get<\/p>\n

\\[ \\mathrm{Var}_\\mu(f) \\leq\\int\\langle(\\nabla^2V)^{-1}\\nabla f,\\nabla f\\rangle\\mathrm{d}\\mu. \\]<\/p>\n

This is the Brascamp-Lieb inequality<\/b>, which can be proved by many ways. If \\( {\\mu} \\) is uniformly log-concave<\/b> : \\( {\\nabla^2V\\geq\\rho\\mathrm{Id}} \\) as quadratic forms for some constant \\( {\\rho>0} \\), then we obtain a Poincar\u00e9 inequality<\/b> of constant \\( {1\/\\rho} \\) :<\/p>\n

\\[ \\mathrm{Var}_\\mu(f) \\leq\\frac{1}{\\rho} \\int\\|\\nabla f\\|^2\\mathrm{d}\\mu =\\frac{1}{\\rho}\\int f(-\\mathrm{L})f\\mathrm{d}\\mu. \\]<\/p>\n

In other words \\( {c_{\\mathrm{P}}\\leq\\frac{1}{\\rho}} \\). Equality is achieved for instance in the gaussian case \\( {C\\equiv0} \\). For Bakry-\u00c9mery connoisseurs, the inequality \\( {(-\\mathrm{L})\\geq\\rho \\mathrm{Id}} \\) on differential forms (gradients) means \\( {\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu\\geq\\rho\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu} \\), which is, thanks to the mean Bochner formula, nothing else but the inequality \\( {\\int\\Gamma_2(f)\\mathrm{d}\\mu\\geq\\rho\\int\\Gamma(f)\\mathrm{d}\\mu} \\) known as the ``crit\u00e8re \\( {\\Gamma_2} \\) int\u00e9gr\u00e9''.<\/p>\n

Apparently, the Brascamp-Lieb inequality was already known by Lars H\u00f6rmander.<\/p>\n

Bakry-\u00c9mery \\( {\\Gamma_2} \\).<\/b> The Bakry-\u00c9mery \\( {\\Gamma_2} \\) is an abstraction of the Bochner commutation-curvature formula. Indeed, having in mind that \\( {\\mathrm{L}=\\Delta-\\langle\\nabla V,\\nabla\\rangle} \\), we find, for all \\( {f,\\varphi} \\),<\/p>\n

\\[ \\mathrm{L}(\\varphi(f)) =\\varphi'(f)\\mathrm{L}f+\\varphi''(f)\\|\\nabla f\\|^2, \\]<\/p>\n

in particular \\( {\\mathrm{L}(f^2)=2f\\mathrm{L}f+2\\|\\nabla f\\|^2} \\), which leads to the functional quadratic form<\/p>\n

\\[ \\Gamma(f,f):=\\tfrac{1}{2}\\mathrm{L}(f^2)-f\\mathrm{L}f=\\|\\nabla f\\|^2, \\]<\/p>\n

and equivalently or more generally,<\/p>\n

\\[ \\Gamma(f,g) =\\tfrac{1}{2}\\mathrm{L}(fg)-f\\mathrm{L}g-g\\mathrm{L}f =\\langle\\nabla f,\\nabla g\\rangle. \\]<\/p>\n

Similarly, we find, using also this time the Bochner commutation-curvature formula,<\/p>\n

\\[ \\mathrm{L}\\Gamma(f,f) =2\\Gamma(f,\\mathrm{L}f) +2(\\|\\nabla^2f\\|_{\\mathrm{HS}}^2+\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle), \\]<\/p>\n

which leads to define<\/p>\n

\\[ \\begin{array}{rcl} \\Gamma_2(f,f) :=\\tfrac{1}{2}\\mathrm{L}\\Gamma(f,f)-\\Gamma(f,\\mathrm{L}f) =\\|\\nabla^2f\\|_{\\mathrm{HS}}^2+\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle. \\end{array} \\]<\/p>\n

At this step, we already know that by averaging over \\( {\\mathrm{d}\\mu(x)=\\mathrm{e}^{-V(x)}\\mathrm{d}x} \\) and using integration by parts, we recover two variants of the mean Bochner formula :<\/p>\n

\\[ \\int\\Gamma_2(f)\\mathrm{d}\\mu =\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu =\\int f\\mathrm{L}^2f\\mathrm{d}\\mu. \\]<\/p>\n

The \\( {\\Gamma} \\) and \\( {\\Gamma_2} \\) objects can be defined on manifolds, in which case a Ricci curvature term appears in \\( {\\Gamma_2} \\), that suggests naturally to also interpret the Hessian \\( {\\nabla^2V} \\) as a curvature. This makes perfectly sense having in mind the Lichnerowicz type comparisons.<\/p>\n

Integrated Bakry-\u00c9mery \\( {\\Gamma_2} \\) criterion.<\/b> It is a characterization of the Poincar\u00e9 inequality that reads as follows, for all \\( {\\rho>0} \\) :<\/p>\n

\\[ \\begin{array}{rcl} &\\displaystyle\\forall f,\\quad \\mathrm{Var}_\\mu(f) \\leq\\frac{1}{\\rho}\\int\\|\\nabla f\\|^2\\mathrm{d}\\mu\\\\ &\\displaystyle\\Updownarrow\\\\ &\\displaystyle\\forall f,\\quad \\int\\|\\nabla f\\|^2\\mathrm{d}\\mu \\leq\\frac{1}{\\rho}\\int(\\|\\nabla f^2\\|_{\\mathrm{HS}}^2+\\langle\\nabla^2V\\nabla f,\\nabla f\\rangle)\\mathrm{d}\\mu, \\end{array} \\]<\/p>\n

in other words, by using the integration by parts and the mean Bochner formula,<\/p>\n

\\[ \\begin{array}{rcl} &\\displaystyle\\forall f\\text{ centered},\\quad \\int f^2\\mathrm{d}\\mu \\leq\\frac{1}{\\rho}\\int f(-\\mathrm{L})f\\mathrm{d}\\mu\\\\ &\\displaystyle\\Updownarrow\\\\ &\\displaystyle \\forall f,\\quad \\int f(-\\mathrm{L})f\\mathrm{d}\\mu \\leq\\frac{1}{\\rho}\\int(\\mathrm{L}f)^2\\mathrm{d}\\mu. \\end{array} \\]<\/p>\n

This equivalence is immediate when using an eigenfunction carrying the spectral gap \\( {\\lambda} \\) and the equivalence of \\( {\\lambda\\geq\\rho} \\) with \\( {c_{\\mathrm{P}}\\leq1\/\\rho} \\). It appears also immediately if we reformulate in terms of functional quadratic forms on centered functions :<\/p>\n

\\[ \\begin{array}{rcl} &\\displaystyle\\mathrm{Id}\\leq\\frac{1}{\\rho}(-\\mathrm{L})\\\\ &\\displaystyle\\Updownarrow\\\\ &\\displaystyle (-\\mathrm{L})\\leq\\frac{1}{\\rho}(-\\mathrm{L})^2. \\end{array} \\]<\/p>\n

This reminds what we did above for the Helffer-Sj\u00f6strand formula, indeed<\/p>\n

\\[ \\int(\\mathrm{L}f)^2\\mathrm{d}\\mu =\\int\\langle\\nabla f,\\nabla(-\\mathrm{L})f\\rangle\\mathrm{d}\\mu. \\]<\/p>\n

Note that on centered functions \\( {\\displaystyle(-\\mathrm{L})^{-1}=\\int_0^\\infty\\mathrm{e}^{t\\mathrm{L}}\\mathrm{d}t} \\), a link to semigroup interpolation.<\/p>\n

Final words.<\/b> The probabilistic and geometric functional analysis contains other comparisons to spheres, for instance the Myers diameter inequality and the L\u00e9vy-Gromov isoperimetric inequality. The analogue comparisons to Gaussians were extensively developed by Dominique Bakry and Michel Ledoux, around a curvature-dimension inequality<\/p>\n

\\[ \\Gamma_2(f)\\geq\\rho\\Gamma(f)+\\frac{1}{n}(\\mathrm{L}f)^2, \\]<\/p>\n

which abstracts the Bochner commutation-curvature formula while incorporating the dimension. For simplicity, this post is free of any semigroup or stochastic process.<\/p>\n

By analogy, we could ask about a Klartag type improvement of the log-Sobolev inequality by incorporating the operator norm of the covariance matrix, something like<\/p>\n

\\[ c_{\\mathrm{P}} \\leq\\frac{c_{\\mathrm{LS}}}{2} \\leq\\sqrt{\\frac{\\|\\Sigma\\|_{\\mathrm{op}}}{\\rho}} \\leq\\rho. \\]<\/p>\n

But it could be something more involved. Regarding this type of analogy, it is already known that there is no log-Sobolev analogue of the Brascamp-Lieb inequality.<\/p>\n

We could ask about the relevance of the Poincar\u00e9 inequality with respect to the spectral gap. Actually, in particular in statistical mechanics, the Poincar\u00e9 inequality is a functional formulation that allows specific methods such as conditioning and the martingale method. It is also related to the quantification of the ergodic phenomenon, and to the geometric analysis related to isoperimetry and concentration of measure. It plays moreover an essential role in the family of Sobolev type inequalities. Depending on your culture or tastes, you may prefer this or that, but a truth is that many aspects are here, connected, waiting for enthusiasm, curiosity, and talent.<\/p>\n

Further reading.<\/b><\/p>\n