Brascamp–Lieb inequality explained

Rn

. It generalizes the Loomis–Whitney inequality and Hölder's inequality. The second is a result of probability theory which gives a concentration inequality for log-concave probability distributions. Both are named after Herm Jan Brascamp and Elliott H. Lieb.

The geometric inequality

Fix natural numbers m and n. For 1 ≤ i ≤ m, let ni ∈ N and let ci > 0 so that

m
\sum
i=1

cini=n.

Choose non-negative, integrable functions

fi\inL1\left(

ni
R

;[0,+infty]\right)

and surjective linear maps

Bi:Rn\to

ni
R

.

Then the following inequality holds:

\int
Rn
m
\prod
i=1

fi\left(Bix

ci
\right)

dx\leqD-

m
\prod
i=1

\left(

\int
ni
R

fi(y)dy

ci
\right)

,

where D is given by

D=inf\left\{\left.

\det\left(
m
\sum
i=1
ci
*
B
i
AiBi\right)
m
\prod(\detAi
ci
)
i=1

\right|Aiisapositive-definiteni x nimatrix\right\}.

Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each

fi

is a centered Gaussian function, namely

fi(y)=\exp\{-(y,Aiy)\}

.[1]

Alternative forms

Consider a probability density function

p(x)=\exp(-\phi(x))

. This probability density function

p(x)

is said to be a log-concave measure if the

\phi(x)

function is convex. Such probability density functions have tails which decay exponentially fast, so most of the probability mass resides in a small region around the mode of

p(x)

. The Brascamp–Lieb inequality gives another characterization of the compactness of

p(x)

by bounding the mean of any statistic

S(x)

.

Formally, let

S(x)

be any derivable function. The Brascamp–Lieb inequality reads:

\operatorname{var}p(S(x))\leqEp(\nablaTS(x)[H\phi(x)]-1\nablaS(x))

where H is the Hessian and

\nabla

is the Nabla symbol.[2]

BCCT inequality

The inequality is generalized in 2008[3] to account for both continuous and discrete cases, and for all linear maps, with precise estimates on the constant.

Definition: the Brascamp-Lieb datum (BL datum)

d,n\geq1

.

d1,...,dn\in\{1,2,...,d\}

.

p1,...,pn\in[0,infty)

.

Bi:\Rd\to

di
\R
are linear surjections, with zero common kernel:

\capiker(Bi)=\{0\}

.

(B,p)=(B1,...,Bn,p1,...,pn)

a Brascamp-Lieb datum (BL datum).

For any

fi\inL1(R

di

)

with

fi\geq0

, defineBL(B, p, f) := \frac

Now define the Brascamp-Lieb constant for the BL datum:BL(B, p) = \max_BL(B, p, f)

Discrete case

Setup:

G,G1,...,Gn

.

\phij:G\toGj

.

(G,G1,...,Gn,\phi1,...\phin)

T(G)

is the torsion subgroup, that is, the subgroup of finite-order elements.

With this setup, we have (Theorem 2.4,[4] Theorem 3.12 [5])

Note that the constant

|T(G)|

is not always tight.

BL polytope

Given BL datum

(B,p)

, the conditions for

BL(B,p)<infty

are

d=\sumipidi

, and

V

of

\Rd

,dim(V) \leq\sum_i p_i dim(B_i(V))

Thus, the subset of

p\in[0,infty)n

that satisfies the above two conditions is a closed convex polytope defined by linear inequalities. This is the BL polytope.

Note that while there are infinitely many possible choices of subspace

V

of

\Rd

, there are only finitely many possible equations of

dim(V)\leq\sumipidim(Bi(V))

, so the subset is a closed convex polytope.

Similarly we can define the BL polytope for the discrete case.

Relationships to other inequalities

The geometric Brascamp–Lieb inequality

The case of the Brascamp - Lieb inequality in which all the ni are equal to 1 was proved earlier than the general case.[6] In 1989, Keith Ball introduced a "geometric form" of this inequality. Suppose that

(ui)

m
i=1
are unit vectors in

Rn

and

(ci)

m
i=1
are positive numbers satisfying
m
\sum
i=1

ci\langlex,ui\rangleui=x

for all

x\inRn

, and that

(fi)

m
i=1
are positive measurable functions on

R

.Then
\int
Rn
m
\prod
i=1

fi(\langlex,ui

ci
\rangle)

dx\leq

m
\prod
i=1

\left(\intRfi(t)dt

ci
\right)

.

Thus, when the vectors

(ui)

resolve the inner product the inequality has a particularly simple form: the constant is equal to 1 and the extremal Gaussian densities are identical. Ball used this inequality to estimate volume ratios and isoperimetric quotients for convex sets in [7] and.[8]

There is also a geometric version of the more general inequality in which the maps

Bi

are orthogonal projections and
m
\sum
i=1

ciBi=I

where

I

is the identity operator on

Rn

.

Hölder's inequality

Take ni = n, Bi = id, the identity map on

Rn

, replacing fi by f, and let ci = 1 / pi for 1 ≤ i ≤ m. Then
m
\sum
i=1
1
pi

=1

and the log-concavity of the determinant of a positive definite matrix implies that D = 1. This yields Hölder's inequality in

Rn

:
\int
Rn
m
\prod
i=1

fi(x)dx\leq

m
\prod
i=1

\|fi

\|
pi

.

Poincaré inequality

The Brascamp–Lieb inequality is an extension of the Poincaré inequality which only concerns Gaussian probability distributions.[9]

Cramér–Rao bound

The Brascamp–Lieb inequality is also related to the Cramér–Rao bound. While Brascamp–Lieb is an upper-bound, the Cramér–Rao bound lower-bounds the variance of

\operatorname{var}p(S(x))

. The Cramér–Rao bound states

\operatorname{var}p(S(x))\geqEp(\nablaTS(x))[Ep(H\phi(x))]-1Ep(\nablaS(x))

.which is very similar to the Brascamp–Lieb inequality in the alternative form shown above.

References

Notes and References

  1. This inequality is in Elliott H.. Lieb. Gaussian Kernels have only Gaussian Maximizers. Inventiones Mathematicae. 102. 179–208. 1990. 10.1007/bf01233426 . free. 1990InMat.102..179L.
  2. This theorem was originally derived in Brascamp . Herm J. . Lieb . Elliott H. . 1976 . On Extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation . . 22 . 4 . 366–389 . 10.1016/0022-1236(76)90004-5 . free. Extensions of the inequality can be found in Hargé . Gilles . 2008 . Reinforcement of an Inequality due to Brascamp and Lieb . . 254 . 2 . 267–300 . 10.1016/j.jfa.2007.07.019 . free. and Carlen . Eric A. . Cordero-Erausquin . Dario . Lieb . Elliott H. . 2013 . Asymmetric Covariance Estimates of Brascamp-Lieb Type and Related Inequalities for Log-concave Measures . Annales de l'Institut Henri Poincaré B . 49 . 1 . 1–12 . 1106.0709 . 2013AIHPB..49....1C . 10.1214/11-aihp462 . free.
  3. Bennett . Jonathan . Carbery . Anthony . Christ . Michael . Tao . Terence . 2008-01-01 . The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals . Geometric and Functional Analysis . en . 17 . 5 . 1343–1415 . 10.1007/s00039-007-0619-6 . 20.500.11820/b13abfca-453c-4aea-adf6-d7d421cec7a4 . 10193995 . 1420-8970. free .
  4. Bennett . Jonathan . Carbery . Anthony . Christ . Michael . Tao . Terence . 2005-05-31 . Finite bounds for Holder-Brascamp-Lieb multilinear inequalities . math/0505691 .
  5. Christ . Michael . Demmel . James . Knight . Nicholas . Scanlon . Thomas . Yelick . Katherine . 2013-07-31 . Communication lower bounds and optimal algorithms for programs that reference arrays -- Part 1 . math.CA . 1308.0068 .
  6. H. J. . Brascamp. E. H. . Lieb. Best Constants in Young's Inequality, Its Converse and Its Generalization to More Than Three Functions. Advances in Mathematics. 20. 2. 151–172. 1976. 10.1016/0001-8708(76)90184-5 . free.
  7. Book: Ball, Keith M. . Keith Martin Ball

    . Keith Martin Ball. Volumes of Sections of Cubes and Related Problems. Geometric Aspects of Functional Analysis. Joram . Lindenstrauss . Joram Lindenstrauss. Vitali D. . Milman . Vitali Milman. Lecture Notes in Mathematics. 1376. 251–260. Springer. Berlin. 1989. 10.1007/BFb0090058 . 978-3-540-51303-2 . free.

  8. Ball. Keith M.. Keith Martin Ball. Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc.. 44. 1991. 351 - 359. 10.1112/jlms/s2-44.2.351. math/9201205.
  9. Log-concavity and strong log-concavity: a review. Saumard . Adrien. Wellner . Jon A.. Statistics Surveys. 8. 45–114. 2014. 10.1214/14-SS107. 27134693 . 4847755 . free.