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 $\mathbb^$. 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

:$\sum_^m c_i n_i = n.$

Choose non-negative, integrable functions

:f_i \in L^1 \left( \mathbb^ ; , + \infty\right)

and surjective linear maps

:$B_i : \mathbb^n \to \mathbb^.$

Then the following inequality holds:

:$\int_ \prod_^m f_i \left( B_i x \right)^ \, \mathrm x \leq D^ \prod_^m \left( \int_ f_i (y) \, \mathrm y \right)^,$

where ''D'' is given by

:$D = \inf \left\.$

Another way to state this is that the constant ''D'' is what one would obtain by restricting attention to the case in which each $f_$ is a centered Gaussian function, namely $f_(y) = \exp \$.

Relationships to other inequalities

The geometric Brascamp–Lieb inequality

The geometric Brascamp–Lieb inequality is a special case of the above, and was used by Keith Ball, in 1989, to provide upper bounds for volumes of central sections of cubes.

For ''i'' = 1, ..., ''m'', let ''c''_''i'' > 0 and let ''u''_''i'' ∈ S^''n''−1 be a unit vector; suppose that ''c''_''i'' and ''u''_''i'' satisfy

:$x = \sum_^m c_i (x \cdot u_i) u_i$

for all ''x'' in R^''n''. Let ''f''_''i'' ∈ ''L''¹(R;  , +∞ for each ''i'' = 1, ..., ''m''. Then

:$\int_ \prod_^m f_i (x \cdot u_i)^ \, \mathrm x \leq \prod_^m \left( \int_ f_i (y) \, \mathrm y \right)^.$

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,

:$B_i^ (z_i) = z_i u_i.$

It follows that ''D'' = 1 in this case.

Hölder's inequality

As another special case, take ''n''_''i'' = ''n'', ''B''_''i'' = id, the identity map on $\mathbb^$, replacing ''f''_''i'' by ''f'', and let ''c''_''i'' = 1 / ''p''_''i'' for 1 ≤ ''i'' ≤ ''m''. Then

:$\sum_^m \frac = 1$

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

:$\int_ \prod_^m f_ (x) \, \mathrm x \leq \prod_^ \, f_i \, _.$

The concentration inequality

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_p (S(x)) \leq E_p (\nabla^T S(x) \phi(x) \nabla S(x))

where H is the Hessian and $\nabla$ is the Nabla symbol.This theorem was originally derived in Extensions of the inequality can be found in and

Relationship with other inequalities

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

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 $\operatorname_p (S(x))$. The expressions are almost identical:

: \operatorname_p (S(x)) \geq E_p (\nabla^T S(x) ) E_p( H \phi(x) ) E_p( \nabla S(x) )\!.

References

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{{DEFAULTSORT:Brascamp-Lieb Inequality
Geometric inequalities