Logarithmic Sobolev inequalities

In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient $\nabla f$. 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:

$\int _{\mathbb {R} ^{n}}{\big |}f(x){\big |}^{2}\log {\big |}f(x){\big |}\,d\nu (x)\leq \int _{\mathbb {R} ^{n}}{\big |}\nabla f(x){\big |}^{2}\,d\nu (x)+\|f\|_{2}^{2}\log \|f\|_{2},$

where $\|f\|_{2}$ 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$$\operatorname {Ent} _{\mu }(f)=\int (f\ln f)d\mu -\int f\ln \left(\int fd\mu \right)d\mu$$This is equal to the (unnormalized) KL divergence by ${\textstyle \operatorname {Ent} _{\mu }(f)=D_{KL}(fd\mu \|(\int fd\mu )d\mu )}$.

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

$\operatorname {Ent} _{\mu }(f^{2})\leq C\int _{\mathbb {R} ^{n}}{\big |}\nabla f(x){\big |}^{2}\,d\mu (x),$

Variants

Lemma ((Tao 2012, Lemma 2.1.16)) — Let ${\textstyle X_{1},\dots ,X_{n}}$ be random variables that are independent, complex-valued, and bounded. ${\textstyle F:\mathbf {C} ^{n}\rightarrow \mathbf {R} }$ be a smooth convex function. Then

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

for some absolute constant ${\textstyle C}$ (independent of ${\textstyle n}$).

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.
• Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics, 97 (4): 1061–1083, doi:10.2307/2373688, JSTOR 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