Nonparametric denoising signals of unknown local structure, II: Nonparametric function recovery

Anatoli B. Juditsky (1) , Arkadii S. Nemirovski (2)

1 SMS - Statistique et Modélisation Stochatisque
2 ISyE - School of Industrial and Systems Engineering [Georgia Tech]

We consider the problem of recovering of continuous multi-dimensional functions from the noisy observations over the regular grid. Our focus is at the adaptive estimation in the case when the function can be well recovered using a linear filter, which can depend on the unknown function itself. In the companion paper "Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities" we have shown in the case when there exists an adapted time-invariant filter, which locally recovers ''well'' the unknown signal, there is a numerically efficient construction of an adaptive filter which recovers the signals ''almost as well''. In the current paper we study the application of the proposed estimation techniques in the non-parametric regression setting. Namely, we propose an adaptive estimation procedure for ''locally well-filtered" signals (some typical examples being smooth signals, modulated smooth signals and harmonic functions) and show that the rate of recovery of such signals in the $\ell_p$-norm on the grid is essentially the same as that rate for regular signals with nonhomogeneous smoothness.

Mots clés: Nonparametric denoising; adaptive filtering; minimax estimation; nonparametric regression

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Anatoli B. Juditsky, Arkadii S. Nemirovski. Nonparametric denoising signals of unknown local structure, II: Nonparametric function recovery. Applied and Computational Harmonic Analysis, 2010, 29 (3), pp.354-367. ⟨10.1016/j.acha.2010.01.003⟩. ⟨hal-00365531⟩