Oracle inequalities for inverse problems

By Laurent Cavalier, G. Golubev, Dominique Picard, Alexandre B. Tsybakov. 2002

204 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7d78153a54988c1f305a673d596f9c1745d23946

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Citations in the corpus

Semi‐Nonparametric IV Estimation of Shape‐Invariant Engel Curves (2003, unknown): “Kress (1999), Korostelev and Tsybakov (1993), Donoho (1995), Engl, et al. (1996), Cavalier, et al. (2002) and Cohen, et al. (2004).”
Comparing parameter choice methods for regularization of ill-posed problems (2011, methodology): “For some methods, there are stronger results involving oracle inequalities [19, 24, 26, 25], which, for continuous white noise, have the form”
Nonparametric statistical inverse problems (2008, methodology): “Then under weak conditions (see [12]) there are constants γ1, γ2 > 0 such that Eθ‖θ − θ‖2 (1 + γ1B−1) min λ∈ R(θ, λ) + γ2Bε (2)[log(DS)] (16) for every θ ∈ 2 and for any B > 0 large enough.”
Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, methodology): “In [10] the data-driven choice of the regularizing parameters is based on unbiased risk estimation.”
Methodology and convergence rates for functional linear regression (2007, background): “[5], but, since the problem is of the same broad type, the rates enjoy the same form and have exactly the same formula if we make the substitutions above.”
Sharp adaptive estimation of the drift function for ergodic diffusions (2005, methodology): “On the other hand, by arguments very similar to those of Lemma 5 in [4], one checks easily that minh∈HN T ES[lT (h)] ∼ minh∈HT ES[lT (h)] ≤ ES[lT (h ∗ T )].”
On the asymptotic properties of the group lasso estimator for linear models (2008, methodology): “Besides ANOVA models, the group-lasso penalty has been applied to generalized linear models in Dahinden et al. (2006), Meier et al.”
Sharp Oracle Inequalities for Aggregation of Affine Estimators (2011, methodology): “• Unbiased risk estimate (URE) minimization with Pinsker’s filters [15]: This method consists in using a Pinsker filter, as defined in Section 6 above, with a datadriven choice of parameters α and w.”
Optimal selection of reduced rank estimators of high-dimensional matrices (2010, background): “See Cavalier et al. [(2002), page 857] for the first claim.”
Risk hull method and regularization by projections of ill-posed inverse problems (2006, unknown): “The Annals of Statistics 2006, Vol. 34, No. 4, 1653–1677 DOI: 10.1214/009053606000000542 © Institute of Mathematical Statistics, 2006”
Oracle Inequalities and Optimal Inference under Group Sparsity (2010, methodology): “The first result is taken from [9], equation (27), and was used in the proof of Lemma 3.”
Structural adaptation via Lp-norm oracle inequalities (2007, methodology): “The following methods can be cited in this context: penalized likelihood estimators, unbiased risk estimators, blockwise Stein estimators, risk hull estimators and so on [see Barron, Birge and Massart (1999), Cavalier et al. (2002), Golubev (2004) and references therein].”
Nonparametric methods for inference in the presence of instrumental variables (2003, background): “Closely related inverse problems, where the context is rendered relatively abstract in order to facilitate solution, include those studied by Donoho [4], Johnstone [8] and Cavalier, Golubev, Picard and Tsybakov [2].”
Estimation of nonparametric conditional moment models with possibly nonsmooth moments (2008, result): “2), our rates coincide with the known minimax optimal rates derived in HH (2005) and Chen and Reiss (2007).(6) In addition, our rates for the general problems of nonlinear and nonsmooth residual functions ρ also coincide with the optimal ones for the linear ill-posed inverse problems.”