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 (y_τ) on the regular grid . 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” {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 ?

– 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.

Keywords: Nonparametric denoising; Oracle inequalities; Adaptive filtering