In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function
f
{\displaystyle f}
, its logarithm, and its gradient
∇
f
{\displaystyle \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:
∫
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},}
where
‖
f
‖
2
{\displaystyle \|f\|_{2}}
is the
L
2
(
ν
)
{\displaystyle L^{2}(\nu )}
-norm of
f
{\displaystyle f}
, with
ν
{\displaystyle \nu }
being standard Gaussian measure on
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 }
This is equal to the (unnormalized) KL divergence by
Ent
μ