Logarithmic Sobolev inequalities

Short description: Class of inequalities

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient $\mathrm{\nabla }f$. These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructive quantum field theory. Similar results were discovered by other mathematicians before and many variations on such inequalities are known.

Gross proved the inequality:

$$\underset{{\mathbb{ℝ}}^n}{\int }|f(x){|}^2\log |f(x)|d\nu (x)\le \underset{{\mathbb{ℝ}}^n}{\int }|\mathrm{\nabla }f(x){|}^{2}d\nu (x)+\Vert f{\Vert }_2^2\log \Vert f{\Vert }_2,$$

where $\Vert f{\Vert }_2$ is the $L^2(\nu )$-norm of $f$, with $\nu$ being standard Gaussian measure on ${\mathbb{ℝ}}^n.$ Unlike classical Sobolev inequalities, Gross's log-Sobolev inequality does not have any dimension-dependent constant, which makes it applicable in the infinite-dimensional limit.

Entropy functional

Define the entropy functional$${\mathrm{Ent}}_{\mu }(f)=\int (f\ln f)d\mu -\int f\ln (\int fd\mu )d\mu$$This is equal to the (unnormalized) KL divergence by ${\mathrm{Ent}}_{\mu }(f)=D_{KL}(fd\mu \Vert (\int fd\mu )d\mu )$.

A probability measure $\mu$ on ${\mathbb{ℝ}}^n$ is said to satisfy the log-Sobolev inequality with constant $C>0$ if for any smooth function f $${\mathrm{Ent}}_{\mu }(f^2)\le C\underset{{\mathbb{ℝ}}^n}{\int }|\mathrm{\nabla }f(x){|}^{2}d\mu (x),$$

Variants

Lemma ((Tao 2012)) — Let $X_1,\dots ,X_n$ be random variables that are independent, complex-valued, and bounded. $F:{\mathbf{𝐂}}^n\to \mathbf{𝐑}$ be a smooth convex function. Then

$$\mathbf{𝐄}F(X)e^{F(X)}\le \left(\mathbf{𝐄}{e}^{F(X)}\right)(\log \mathbf{𝐄}{e}^{F(X)})+C\mathbf{𝐄}{e}^{F(X)}|\mathrm{\nabla }F(X){|}^2$$

for some absolute constant $C$ (independent of $n$).

References

• Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1. 
• Blower, G. (2009). Random matrices: high dimensional phenomena. London Mathematical Society lecture note series. Cambridge, New York: Cambridge University Press. ISBN 978-0-521-13312-8. 
• Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics 97 (4): 1061–1083, doi:10.2307/2373688 
• Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form", Duke Mathematical Journal 42 (3): 383–396, doi:10.1215/S0012-7094-75-04237-4, https://projecteuclid.org/euclid.dmj/1077311187