Logarithmic Sobolev inequalities

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In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f {\displaystyle f} {\displaystyle f}, its logarithm, and its gradient ∇ f {\displaystyle \nabla f} {\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:

∫ R n | f ( x ) | 2 log ⁡ | f ( x ) | d ν ( x ) ≤ ∫ R n | ∇ f ( x ) | 2 d ν ( x ) + ‖ f ‖ 2 2 log ⁡ ‖ f ‖ 2 , {\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},} {\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 ‖ f ‖ 2 {\displaystyle \|f\|_{2}} {\displaystyle \|f\|_{2}} is the L 2 ( ν ) {\displaystyle L^{2}(\nu )} {\displaystyle L^{2}(\nu )}-norm of f {\displaystyle f} {\displaystyle f}, with ν {\displaystyle \nu } {\displaystyle \nu } being standard Gaussian measure on R n . {\displaystyle \mathbb {R} ^{n}.} {\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 Ent μ ⁡ ( f ) = ∫ ( f ln ⁡ f ) d μ − ∫ f ln ⁡ ( ∫ f d μ ) d μ {\displaystyle \operatorname {Ent} _{\mu }(f)=\int (f\ln f)d\mu -\int f\ln \left(\int fd\mu \right)d\mu } {\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 Ent μ ⁡ ( f ) = D K L ( f d μ ‖ ( ∫ f d μ ) d μ ) {\textstyle \operatorname {Ent} _{\mu }(f)=D_{KL}(fd\mu \|(\int fd\mu )d\mu )} {\textstyle \operatorname {Ent} _{\mu }(f)=D_{KL}(fd\mu \|(\int fd\mu )d\mu )}.

A probability measure μ {\displaystyle \mu } {\displaystyle \mu } on R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} is said to satisfy the log-Sobolev inequality with constant C > 0 {\displaystyle C>0} {\displaystyle C>0} if for any smooth function f Ent μ ⁡ ( f 2 ) ≤ C ∫ R n | ∇ f ( x ) | 2 d μ ( x ) , {\displaystyle \operatorname {Ent} _{\mu }(f^{2})\leq C\int _{\mathbb {R} ^{n}}{\big |}\nabla f(x){\big |}^{2}\,d\mu (x),} {\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 X 1 , … , X n {\textstyle X_{1},\dots ,X_{n}} {\textstyle X_{1},\dots ,X_{n}} be random variables that are independent, complex-valued, and bounded. F : C n → R {\textstyle F:\mathbf {C} ^{n}\rightarrow \mathbf {R} } {\textstyle F:\mathbf {C} ^{n}\rightarrow \mathbf {R} } be a smooth convex function. Then

E F ( X ) e F ( X ) ≤ ( E e F ( X ) ) ( log ⁡ E e F ( X ) ) + C E e F ( X ) | ∇ F ( X ) | 2 {\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}} {\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 C {\textstyle C} {\textstyle C} (independent of n {\textstyle n} {\textstyle n}).

Notes

References