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Brascamp–Lieb inequality
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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 . 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.
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The geometric inequality
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Fix natural numbers m and n. For 1 ≤ i ≤ m, let ni ∈ N and let ci > 0 so that
Choose non-negative, integrable functions
Then the following inequality holds:
where D is given by
Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each is a centered Gaussian function, namely .[1]
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Alternative forms
Consider a probability density function . This probability density function is said to be a log-concave measure if the 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 . The Brascamp–Lieb inequality gives another characterization of the compactness of by bounding the mean of any statistic .
Formally, let be any derivable function. The Brascamp–Lieb inequality reads:
where H is the Hessian and is the Nabla symbol.[2]
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BCCT inequality
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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)
- .
- .
- .
- are linear surjections, with zero common kernel: .
- Call a Brascamp-Lieb datum (BL datum).
For any with , define
Now define the Brascamp-Lieb constant for the BL datum:
Theorem—(BCCT, 2007)
is finite iff , and for all subspace of ,
is reached by gaussians:
- If is finite, then there exists some linear operators such that achieves the upper bound.
- If is infinite, then there exists a sequence of gaussians for which
Discrete case
Setup:
- BL datum defined as
- 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])
Theorem—If there exists some such that
Then for all ,
and in particular,
Note that the constant is not always tight.
BL polytope
Given BL datum , the conditions for are
- , and
- for all subspace of ,
Thus, the subset of 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 of , there are only finitely many possible equations of , so the subset is a closed convex polytope.
Similarly we can define the BL polytope for the discrete case.
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Relationships to other inequalities
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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 are unit vectors in and are positive numbers satisfying
for all , and that are positive measurable functions on . Then
Thus, when the vectors 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 are orthogonal projections and
where is the identity operator on .
Hölder's inequality
Take ni = n, Bi = id, the identity map on , replacing fi by f1/ci
i, and let ci = 1 / pi for 1 ≤ i ≤ m. Then
and the log-concavity of the determinant of a positive definite matrix implies that D = 1. This yields Hölder's inequality in :
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.[9] While Brascamp–Lieb is an upper-bound, the Cramér–Rao bound lower-bounds the variance of . The Cramér–Rao bound states
- .
which is very similar to the Brascamp–Lieb inequality in the alternative form shown above.
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References
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