In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient
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: where
( ν )
ν
being standard Gaussian measure on
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.
Define the entropy functional
Ent
( f ) = ∫ ( f ln f ) d μ − ∫ f ln
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
is said to satisfy the log-Sobolev inequality with constant
if for any smooth function f Lemma ((Tao 2012, Lemma 2.1.16)) — Let
be random variables that are independent, complex-valued, and bounded.
be a smooth convex function.
for some absolute constant