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 Z d. 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 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 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.