Adaptive estimation of a quadratic functional of a density by model selection

Laurent, Béatrice

doi:10.1051/ps:2005001

Cited by 169 publications

(268 citation statements)

References 17 publications

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“…In this section we show how Theorem 1.1' can be refined using the following set of estimates for a chi-square random variable: Fact 4.1 (Laurent and Massart [8]) Let ξ 1 , . .…”

Section: A Refinement Of the Main Resultsmentioning

confidence: 99%

A Unified Theorem on SDP Rank Reduction

Zhang

2008

Mathematics of OR

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We consider the problem of finding a low-rank approximate solution to a system of linear equations in symmetric, positive semidefinite matrices, where the approximation quality of a solution is measured by its maximum relative deviation, both above and below, from the prescribed quantities. We show that a simple randomized polynomialtime procedure produces a low-rank solution that has provably good approximation qualities. Our result provides a unified treatment of and generalizes several well-known results in the literature. In particular, it contains as special cases the Johnson-Lindenstrauss lemma on dimensionality reduction, results on low-distortion embeddings into low-dimensional Euclidean space, and approximation results on certain quadratic optimization problems.

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“…In this section we show how Theorem 1.1' can be refined using the following set of estimates for a chi-square random variable: Fact 4.1 (Laurent and Massart [8]) Let ξ 1 , . .…”

Section: A Refinement Of the Main Resultsmentioning

confidence: 99%

A Unified Theorem on SDP Rank Reduction

Zhang

2008

Mathematics of OR

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“…, f(1)) T . As in Theorem 2 from Baraud [1], by using some deviations inequalities due to Birgé [3] and Laurent and Massart [13], since the dimension of S D * k is smaller than D * k , we can prove that for all f satisfying: (6). Let f be some periodic function with period k/n such that for all…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…Given some levels α and δ ∈]0, 1[, let T α be the test statistic defined by (9) with M and {S m , m ∈ M} chosen as in (13) and (14). Introduce for s ∈ N * , R > 0 and …”

Section: Propositionmentioning

confidence: 99%

“…It achieves the parametric rate of testing over directional alternatives. The test developed by Baraud, Huet and Laurent in [2] is based on a criterion related to the one used by Laurent and Massart in [13] to estimate quadratic functionals in the regression framework. It consists in a multiple testing procedure which can be described as follows.…”

Section: ρ(S α δ σ) = Infmentioning

confidence: 99%

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Adaptive tests for periodic signal detection with applications to laser vibrometry

Fromont

Lévy‐Leduc

2006

ESAIM: PS

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Abstract.Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2−distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view.Mathematics Subject Classification. 62G10, 62G08, 62G20.

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“…For the initial case t = 2, the deterministic algorithm by Alon and Naor [1] (Step 3 of Algorithm B 1 ) guarantees a constant ratio, i.e., (15). Suppose now (15) holds for t − 1.…”

Section: Repeat the Above Procedures φmentioning

confidence: 99%

Probability Bounds for Polynomial Functions in Random Variables

Jiang

et al. 2014

Mathematics of OR

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Random sampling is a simple but powerful method in statistics and the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ R n . To do so, one may select a simpler (even finite) subset S 0 ⊆ S, randomly take some samples over S 0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S 0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξ i (1 ≤ i ≤ n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob f (ξ 1 , ξ 2 , · · · , ξ n ) ≥ c 1 n − d 2 F 1 ≥ c 2 , where c 1 , c 2 > 0 are two universal constants. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.

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Adaptive estimation of a quadratic functional of a density by model selection

Cited by 169 publications

References 17 publications

A Unified Theorem on SDP Rank Reduction

A Unified Theorem on SDP Rank Reduction

Adaptive tests for periodic signal detection with applications to laser vibrometry

Probability Bounds for Polynomial Functions in Random Variables

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