Brascamp–Lieb inequality

In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. 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 n_i ∈ N and let c_i > 0 so that

${\displaystyle \sum _{i=1}^{m}c_{i}n_{i}=n.}$

Choose non-negative, integrable functions

${\displaystyle f_{i}\in L^{1}\left(\mathbb {R} ^{n_{i}};[0,+\infty ]\right)}$

and surjective linear maps

${\displaystyle B_{i}:\mathbb {R} ^{n}\to \mathbb {R} ^{n_{i}}.}$

Then the following inequality holds:

${\displaystyle \int _{\mathbb {R} ^{n}}\prod _{i=1}^{m}f_{i}\left(B_{i}x\right)^{c_{i}}\,\mathrm {d} x\leq D^{-1/2}\prod _{i=1}^{m}\left(\int _{\mathbb {R} ^{n_{i}}}f_{i}(y)\,\mathrm {d} y\right)^{c_{i}},}$

where D is given by

${\displaystyle D=\inf \left\{\left.{\frac {\det \left(\sum _{i=1}^{m}c_{i}B_{i}^{*}A_{i}B_{i}\right)}{\prod _{i=1}^{m}(\det A_{i})^{c_{i}}}}\right|A_{i}{\text{ is a positive-definite }}n_{i}\times n_{i}{\text{ matrix}}\right\}.}$

Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each ${\displaystyle f_{i}}$ is a centered Gaussian function, namely ${\displaystyle f_{i}(y)=\exp\{-(y,\,A_{i}\,y)\}}$.^[1]

Alternative forms

Consider a probability density function ${\displaystyle p(x)=\exp(-\phi (x))}$. This probability density function ${\displaystyle p(x)}$ is said to be a log-concave measure if the ${\displaystyle \phi (x)}$ 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 ${\displaystyle p(x)}$. The Brascamp–Lieb inequality gives another characterization of the compactness of ${\displaystyle p(x)}$ by bounding the mean of any statistic ${\displaystyle S(x)}$.

Formally, let ${\displaystyle S(x)}$ be any derivable function. The Brascamp–Lieb inequality reads:

${\displaystyle \operatorname {var} _{p}(S(x))\leq E_{p}(\nabla ^{T}S(x)[H\phi (x)]^{-1}\nabla S(x))}$

where H is the Hessian and ${\displaystyle \nabla }$ 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)

• ${\displaystyle d,n\geq 1}$.

• ${\displaystyle d_{1},...,d_{n}\in \{1,2,...,d\}}$.

• ${\displaystyle p_{1},...,p_{n}\in [0,\infty )}$.

• ${\displaystyle B_{i}:\mathbb {R} ^{d}\to \mathbb {R} ^{d_{i}}}$ are linear surjections, with zero common kernel: ${\displaystyle \cap _{i}ker(B_{i})=\{0\}}$.
• Call ${\displaystyle (B,p)=(B_{1},...,B_{n},p_{1},...,p_{n})}$ a Brascamp-Lieb datum (BL datum).

For any ${\displaystyle f_{i}\in L^{1}(R^{d_{i}})}$ with ${\displaystyle f_{i}\geq 0}$, define

Now define the Brascamp-Lieb constant for the BL datum:

Discrete case

Setup:

• Finitely generated abelian groups ${\displaystyle G,G_{1},...,G_{n}}$.

• Group homomorphisms ${\displaystyle \phi _{j}:G\to G_{j}}$.

• BL datum defined as ${\displaystyle (G,G_{1},...,G_{n},\phi _{1},...\phi _{n})}$

• ${\displaystyle T(G)}$ 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 ${\displaystyle |T(G)|}$ is not always tight.

BL polytope

Given BL datum ${\displaystyle (B,p)}$, the conditions for ${\displaystyle BL(B,p)<\infty }$ are

• ${\displaystyle d=\sum _{i}p_{i}d_{i}}$, and
• for all subspace ${\displaystyle V}$ of ${\displaystyle \mathbb {R} ^{d}}$,
  $${\displaystyle dim(V)\leq \sum _{i}p_{i}dim(B_{i}(V))}$$

  

Thus, the subset of ${\displaystyle p\in [0,\infty )^{n}}$ 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 ${\displaystyle V}$ of ${\displaystyle \mathbb {R} ^{d}}$, there are only finitely many possible equations of ${\displaystyle dim(V)\leq \sum _{i}p_{i}dim(B_{i}(V))}$, 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 geometric Brascamp–Lieb inequality, first derived in 1976,^[6] is a special case of the general inequality. It was used by Keith Ball, in 1989, to provide upper bounds for volumes of central sections of cubes.^[7]

For i = 1, ..., m, let c_i > 0 and let u_i ∈ Sⁿ⁻¹ be a unit vector; suppose that c_i and u_i satisfy

${\displaystyle x=\sum _{i=1}^{m}c_{i}(x\cdot u_{i})u_{i}}$

for all x in Rⁿ. Let f_i ∈ L¹(R; [0, +∞]) for each i = 1, ..., m. Then

${\displaystyle \int _{\mathbb {R} ^{n}}\prod _{i=1}^{m}f_{i}(x\cdot u_{i})^{c_{i}}\,\mathrm {d} x\leq \prod _{i=1}^{m}\left(\int _{\mathbb {R} }f_{i}(y)\,\mathrm {d} y\right)^{c_{i}}.}$

The geometric Brascamp–Lieb inequality follows from the Brascamp–Lieb inequality as stated above by taking n_i = 1 and B_i(x) = x · u_i. Then, for z_i ∈ R,

${\displaystyle B_{i}^{*}(z_{i})=z_{i}u_{i}.}$

It follows that D = 1 in this case.

Hölder's inequality

Take ni = n, Bi = id, the identity map on ${\displaystyle \mathbb {R} ^{n}}$, replacing fi by f1/ci
i, and let ci = 1 / pi for 1 ≤ i ≤ m. Then

${\displaystyle \sum _{i=1}^{m}{\frac {1}{p_{i}}}=1}$

and the log-concavity of the determinant of a positive definite matrix implies that D = 1. This yields Hölder's inequality in ${\displaystyle \mathbb {R} ^{n}}$:

${\displaystyle \int _{\mathbb {R} ^{n}}\prod _{i=1}^{m}f_{i}(x)\,\mathrm {d} x\leq \prod _{i=1}^{m}\|f_{i}\|_{p_{i}}.}$

Poincaré inequality

The Brascamp–Lieb inequality is an extension of the Poincaré inequality which only concerns Gaussian probability distributions.^[8]

Cramér–Rao bound

The Brascamp–Lieb inequality is also related to the Cramér–Rao bound.^[8] While Brascamp–Lieb is an upper-bound, the Cramér–Rao bound lower-bounds the variance of ${\displaystyle \operatorname {var} _{p}(S(x))}$. The Cramér–Rao bound states

${\displaystyle \operatorname {var} _{p}(S(x))\geq E_{p}(\nabla ^{T}S(x))[E_{p}(H\phi (x))]^{-1}E_{p}(\nabla S(x))\!}$.

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

References

• Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bulletin of the American Mathematical Society. New Series. 39 (3): 355–405. doi:10.1090/S0273-0979-02-00941-2.