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

TL;DR

This paper develops an adaptive, nonparametric method for pointwise estimation of multi-dimensional signals from noisy data, focusing on signals that can be well approximated by unknown linear filters, and demonstrates the method's effectiveness and the richness of such signals.

Contribution

It introduces a numerically efficient adaptive estimator for signals well-filtered by unknown linear filters, expanding the class of signals that can be effectively recovered from noisy observations.

Findings

The estimator achieves near-oracle performance in denoising.

The class of well-filtered signals includes smooth, modulated, and harmonic functions.

The proposed method is computationally feasible for high-dimensional data.

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 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.