Nonparametric denoising of signals with unknown local structure, I: Oracle inequalities

We consider the problem of pointwise estimation of multi-dimensional signals s, from noisy observations (yτ) on the regular grid Zd. 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 with the coefficients which vanish outside the “cube” d{0,…,T} which recovers s0 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 s0, which relies upon observations and recovers s0 with basically the same estimation error as the unknown filter ?–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.