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 ${\displaystyle \nabla f}$. These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form,^[1][2] 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^[3] proved the inequality:

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

where ${\displaystyle \|f\|_{2}}$ is the ${\displaystyle L^{2}(\nu )}$-norm of ${\displaystyle f}$, with ${\displaystyle \nu }$ being standard Gaussian measure on ${\displaystyle \mathbb {R} ^{n}.}$ 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$${\displaystyle \operatorname {Ent} _{\mu }(f)=\int (f\ln f)d\mu -\int f\ln \left(\int fd\mu \right)d\mu }$$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 ${\displaystyle \mu }$ on ${\displaystyle \mathbb {R} ^{n}}$ is said to satisfy the log-Sobolev inequality with constant ${\displaystyle C>0}$ if for any smooth function f $${\displaystyle \operatorname {Ent} _{\mu }(f^{2})\leq C\int _{\mathbb {R} ^{n}}{\big |}\nabla f(x){\big |}^{2}\,d\mu (x),}$$

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

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

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