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Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities

Abstract : We consider the problem of pointwise estimation of multi-dimensional signals $s$, from noisy observations $(y_\tau)$ on the regular grid $\bZd$. Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. \par The basic setting of the problem we address here can be summarized as follows: suppose that the signal $s$ is ''well-filtered'', i.e. there exists an adapted time-invariant linear filter $q^*_T$ with the coefficients which vanish outside the ''cube'' $\{0,..., T\}^d$ which recovers $s_0$ from observations with small mean-squared error. We suppose that we do not know the filter $q^*$, although, we do know that such a filter exists. We give partial answers to the following questions:
  • is it possible to construct an adaptive estimator of the value $s_0$, which relies upon observations and recovers $s_0$ with basically the same estimation error as the unknown filter $q^*_T$?
  • how rich is the family of well-filtered (in the above sense) signals?
We show that the answer to the first question is affirmative and provide a numerically efficient construction of a nonlinear adaptive filter. Further, we establish a simple calculus of ''well-filtered" signals, and show that their family is quite large: it contains, for instance, sampled smooth signals, sampled modulated smooth signals and sampled harmonic functions.
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Submitted on : Wednesday, September 3, 2008 - 6:24:58 PM
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Anatoli Juditsky, Arkadii S. Nemirovski. Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities. Applied and Computational Harmonic Analysis, Elsevier, 2009, 27 (2), pp.157-179. ⟨10.1016/j.acha.2009.02.001⟩. ⟨hal-00318084⟩



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