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- W2950458818 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. 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 (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." @default.
- W2950458818 created "2019-06-27" @default.
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- W2950458818 date "2008-09-04" @default.
- W2950458818 modified "2023-09-27" @default.
- W2950458818 title "Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities" @default.
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