Approximate Independence of Distributions on Spheres and Their Stability Properties

Rachev, Svetlozar T.; Rüschendorf, Ludger

doi:10.1214/aop/1176990346

Cited by 74 publications

(84 citation statements)

References 13 publications

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“…Taking α = (2p − 2)/p, we obtain the required bounds on n i=1 y α i = n i=1 |x i | 2p−2 . The proofs of our results rely on the following probabilistic representation for the cone probability measure on S n−1 p for p ∈ [1, ∞) (and for the uniform distribution over B n p ), which is due to Schechtman and Zinn [17] and was independently obtained by Rachev and Rüschendorf in [16]. with respect to the Lebesgue measure on R. Then, for Z := (Z 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

Yet another note on the arithmetic-geometric mean inequality

2020

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It was shown by E. Gluskin and V.D. Milman in [GAFA Lecture Notes in Math. 1807, 2003] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to coordinates of a point chosen with respect to the surface unit measure on a high-dimensional Euclidean sphere. We present here two asymptotic refinements of this phenomenon in the more general setting of the surface probability measure on a high-dimensional p -sphere, and also show that sampling the point according to either the cone probability measure on p or the uniform distribution on the ball enclosed by p yields the same results. First, we prove a central limit theorem, which allows us to identify the precise constants in the reverse inequality. Second, we prove the large deviations counterpart to the central limit theorem, thereby describing the asymptotic behavior beyond the Gaussian scale, and identify the rate function.

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Section: Preliminariesmentioning

confidence: 99%

Yet another note on the arithmetic-geometric mean inequality

2020

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“…( [RR91,SZ90]; see also [BGMN05], Theorem 2) Let Xi ∼ 1 √ π e −|x i | 2 . Then the random vector X1 (|X1| 2 + .…”

Section: Proofmentioning

confidence: 99%

Smooth Tradeoffs between Insert and Query Complexity in Nearest Neighbor Search

Kapralov

2015

Proceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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Locality Sensitive Hashing (LSH) has emerged as the method of choice for high dimensional similarity search, a classical problem of interest in numerous applications. LSH-based solutions require that each data point be inserted into a number A of hash tables, after which a query can be answered by performing B lookups. The original LSH solution of [IM98] showed for the first time that both A and B can be made sublinear in the number of data points. Unfortunately, the classical LSH solution does not provide any tradeoff between insert and query complexity, whereas for data (respectively, query) intensive applications one would like to minimize insert time by choosing a smaller A (respectively, minimize query time by choosing a smaller B). A partial remedy for this is provided by Entropy LSH [Pan06], which allows to make either inserts or queries essentially constant time at the expense of a loss in the other parameter, but no algorithm that achieves a smooth tradeoff is known.In this paper, we present an algorithm for performing similarity search under the Euclidean metric that resolves the problem above. Our solution is inspired by Entropy LSH, but uses a very different analysis to achieve a smooth tradeoff between insert and query complexity. Our results improve upon or match, up to lower order terms in the exponent, best known data-oblivious algorithms for the Euclidean metric.

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“…We shall also use the following polar integration formula, stated here only for the case of ℓ d p -balls: [24]) that provides a probabilistic representation of the cone measure m B d p of the unit ball of ℓ d p (see also [4] for an extension) and shall be used later with the previous polar integration formula. Proposition 2.9.…”

Section: 25mentioning

confidence: 99%

The curse of dimensionality for numerical integration on general domains

Hinrichs

Prochno

Ullrich

2019

Journal of Complexity

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We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal ψ 2 -estimate. In particular, we obtain the result for the important class of volume-normalized ℓ d p -balls in the complete regime 2 ≤ p ≤ ∞. This extends a result in a work of A. Hinrichs, E. Novak, M. Ullrich and H. Woźniakowski [13] to the whole range 2 ≤ p ≤ ∞, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Guédon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.

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Approximate Independence of Distributions on Spheres and Their Stability Properties

Cited by 74 publications

References 13 publications

Yet another note on the arithmetic-geometric mean inequality

Yet another note on the arithmetic-geometric mean inequality

Smooth Tradeoffs between Insert and Query Complexity in Nearest Neighbor Search

The curse of dimensionality for numerical integration on general domains

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