Decoupling and Khintchine's Inequalities for $U$-Statistics

Peña, Víctor H. de la

doi:10.1214/aop/1176989533

Cited by 61 publications

(27 citation statements)

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“…, A N be an orthonormal basis for the linear space of n × n matrices with the usual inner product X, Y = trace(X * Y ). Then we expect our results should also apply to the rank minimization problem 14) where ⊂ {1, . .…”

Section: Extensionsmentioning

confidence: 91%

“…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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Section: Extensionsmentioning

confidence: 91%

Section: Improvementsmentioning

confidence: 99%

Exact Matrix Completion via Convex Optimization

2009

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“…We consider the U -statistics (without any normalisation) U (2) n (f ) = Σ 1≤i =j≤n f (ξ i , ξ j ) · U (1) n (π 1 f ) = Σ 1≤i≤n π 1 f (ξ i ) · U (2) n (π 2 f ) = Σ 1≤i =j≤n π 2 f (ξ i , ξ j ), thus U (2) n (f − Ef )) = 2(n − 1)U (1) n (π 1 f ) + U (2) n (π 2 f ) ·…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…R-valued random variables) as the main part and it can be interesting to bound separately the second part U (2) n (π 2 f ). Now, as π 2 f is canonical of order 2, if |π 2 f | ≤ c Keywords and phrases: Large deviations, U-statistics.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…and σ 2 = E(π 2 f ) 2 , there exist two constants c 1 , c 2 such that P (|U (2) n (π 2 f )| ≥ (n − 1)t) ≤ c 1 exp(−c 2 t/{σ + c 2/3 t 1/3 n −1/3 }) ( A G 2 ) ((AG1) and (AG2) can be found in Arcones and Giné [1]). The normalized version of inequality (AG2) can be restated P (|U (2) n (π 2 f )|) ≥ a 3 Max (nσx, c…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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