ADAPTIVE ESTIMATION OF A QUADRATIC FUNCTIONAL BY MODEL SELECTION

By Béatrice Laurent, Pascal Massart. 2001

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Citations in the corpus

Adversarial Examples Are Not Bugs, They Are Features (2019, neutral): “Note, however, that via the result of Laurent and Massart [LM00] showing strong concentration for the norms of Gaussian random variables, in high dimensions this bound on expectation has a corresponding high-probability bound on the norm.”
Sharpness-Aware Minimization for Efficiently Improving Generalization (2020, positive): “Therefore, 22 has chi-square distribution and by Lemma 1 in Laurent & Massart (2000), we have that for any positive t : Therefore, with probability 1 − 1 / √ n we have that: Substituting the above value for σ back to the inequality and using theorem’s assumption gives us following inequality:”
A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks (2018, background): “The following lemma — a direct consequence of the Chernoff bound applied to the χ 2 distribution with d 0 degrees of freedom — will give us the desired result: Lemma 9 (Laurent and Massart (2000), Lemma 1) .”
Exact Matrix Completion via Convex Optimization (2008, methodology): “if Cof the form Kwhere Kis a xed numerical constant, we can achieve a probability at least 1 cn provided that nis suciently large. To establish these facts, we make use of the standard result below [21]. Lemma 2.3 Let Y dbe distributed as a chi-squared random variable with ddegrees of freedom. Then ”
On the Convergence Rate of Training Recurrent Neural Networks (2018, background): “3 (Lemma 1 on page 1325 of Laurent and Massart [35]).”
Robust Regression via Hard Thresholding (2015, methodology): “[15] Beatrice Laurent and Pascal Massart.”
Privacy via the Johnson-Lindenstrauss Transform (2012, background): “(by [21, Lemma 1]) Using the union bound and plugging in the bound on Z 1 into the second expression, we obtain the desired bound.”
Are Labels Required for Improving Adversarial Robustness? (2019, methodology): “The result follows from application of Lemma 1 from [23].”
Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming (2010, background): “(ǫ2 −1) < −rn} = pr{ Pn i=1(ǫ 2 i −1) < −nrn} = pr{ Pn i=1 ai(ǫ 2 i −1) < −2|a|2 √ nrn/2} where we have ai = 1 (i = 1,...,n), so that |a|2 = √ n. Applying the second inequality of Lemma 1 of [12] for √ x = √ nrn/2, we have pr{En(ǫ2) < (1 −rn)2} 6 exp(−nr2 n/4). E.2. Omitted Part of Proof”
Robust Lasso With Missing and Grossly Corrupted Observations (2011, methodology): “The following tail bounds on the Chi-square variates taken from [47] are useful”
All-but-the-Top: Simple and Effective Postprocessing for Word Representations (2017, background): “We divide the proof into the following steps: • C follows chi-square distribution with the degree of freedom of d , thus C can be bounded by (Laurent and Massart, 2000), Combining the two inequalities above, one has (8) proved.”
Model selection via testing: an alternative to (penalized) maximum likelihood estimators (2006, background): “Since a non-central χ2 variable is stochastically larger than a central one, it follows from Lemma 1 of Laurent and Massart [40] that”
On the Randomized Complexity of Minimizing a Convex Quadratic Function (2018, result): “This can in turn be controlled by recent a result due to Lelarge and Miolane [2016], which rigorously establishes a general replica-symmetric formula for planted matrix models.”
Maximum likelihood estimation for linear Gaussian covariance models (2014, methodology): “’s inequality we nd that (A.1) P jTr(W n) npj 1 2np 2 : Finite-sample bounds that are more accurate than (A.1) can be obtained using results in [IL06,Ing10,LM00]. We present here the bounds from [LM00]. Let X˘˜2 d ; using the original formulation in [LM00, page 1325], we obtain for any positive x, ”
Adaptive nonparametric confidence sets (2006, background): “By James Robins and Aad van der Vaart”