Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics

Peña, Víctor H. de la; Montgomery‐Smith, Stephen

doi:10.1214/aop/1176988291

Cited by 82 publications

(61 citation statements)

References 6 publications

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“…, T ′ n are independent random variables. The decoupling inequality of de la Peña and Montgomery-Smith [2] states that, whenever f i,i = 0 and f i,j = f j,i for all i, j,…”

Section: Discussionmentioning

confidence: 99%

Exponential Inequalities, with Constants, for U-statistics of Order Two

Houdré

Reynaud-Bouret²

2003

Stochastic Inequalities and Applications

103

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“…, T ′ n are independent random variables. The decoupling inequality of de la Peña and Montgomery-Smith [2] states that, whenever f i,i = 0 and f i,j = f j,i for all i, j,…”

Section: Discussionmentioning

confidence: 99%

Exponential Inequalities, with Constants, for U-statistics of Order Two

Houdré

Reynaud-Bouret²

2003

Stochastic Inequalities and Applications

103

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“…This paper generalizes the notion of incoherence to problems beyond the setting of sparse signal recovery. In [29], the authors studied the nuclear norm heuristic applied to a related problem where partial information about a matrix M is available from m equations of the form 15) where for each k, {A…”

Section: Connections Alternatives and Prior Artmentioning

confidence: 99%

“…Lemma 6.5 [15] Let {η i } 1≤i≤n be a sequence of independent random variables, and {x ij } i =j be elements taken from a Banach space. Then…”

Section: Lemma 64mentioning

confidence: 99%

“…To obtain an optimal result, one would need to reach k of size about log n. In doing so, however, one would have to pay special attention to the size of the decoupling constants (the constant C D for two variables in Lemma 6.5) which depend on k-the number of decoupled variables. These constants grow with k and upper bounds are known [14,15].…”

Section: Improvementsmentioning

confidence: 99%

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Exact Matrix Completion via Convex Optimization

2009

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We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen?We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys m ≥ C n 1.2 r log n for some positive numerical constant C, then with very high probability, most n × n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.Communicated by Michael Todd.E.J. Candès ( ) Applied and Computational Mathematics,

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“…Thanks to general decoupling inequalities for U -statistics [25], which we recall in the "Appendix" (Theorem 7.1), the above theorem is formally equivalent to Theorem 1.1. In fact in [44] Latała first proves the above version.…”

Section: A Concentration Inequality For Non-lipschitz Functionsmentioning

confidence: 99%

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

Adamczak

Wolff

2014

Probab. Theory Relat. Fields

163

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Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for not necessarily Lipschitz functions f : R n → R with bounded derivatives of higher orders, which holds when the underlying measure satisfies a family of Sobolev type inequalitiesSuch Sobolev type inequalities hold, e.g., if the underlying measure satisfies the logSobolev inequality (in which case C( p) ≤ C √ p) or the Poincaré inequality (then C( p) ≤ C p). Our concentration estimates are expressed in terms of tensor-product norms of the derivatives of f . When the underlying measure is Gaussian and f is a polynomial (not necessarily tetrahedral or homogeneous), our estimates can be reversed (up to a constant depending only on the degree of the polynomial). We also show that for polynomial functions, analogous estimates hold for arbitrary random vectors with independent sub-Gaussian coordinates. We apply our inequalities to general additive functionals of random vectors (in particular linear eigenvalue statistics of random matrices) and the problem of counting cycles of fixed length in Erdős-Rényi random graphs, obtaining new estimates, optimal in a certain range of parameters.

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Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics

Cited by 82 publications

References 6 publications

Exponential Inequalities, with Constants, for U-statistics of Order Two

Exponential Inequalities, with Constants, for U-statistics of Order Two

Exact Matrix Completion via Convex Optimization

Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order

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