Adaptive Denoising of Signals with Local Shift-Invariant Structure

TL;DR

This paper introduces a new adaptive signal denoising method based on minimizing the residual's -norm, offering better statistical properties and bounds under the assumption of approximate shift-invariance.

Contribution

It proposes a novel -norm based adaptive estimator for signals with local shift-invariant structure, improving upon previous -infinity based methods.

Findings

The -norm residual minimization estimator has superior statistical properties.

Derived oracle inequalities for - loss and bounds for  and pointwise errors.

Application to denoising harmonic oscillations.

Abstract

We discuss the problem of adaptive discrete-time signal denoising in the situation where the signal to be recovered admits a "linear oracle" -- an unknown linear estimate that takes the form of convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski (2009) that when the $\ell_2$-norm of the oracle filter is small enough, such oracle can be "mimicked" by an efficiently computable adaptive estimate of the same structure with an observation-driven filter. The filter in question was obtained as a solution to the optimization problem in which the $\ell_\infty$-norm of the Discrete Fourier Transform (DFT) of the estimation residual is minimized under constraint on the $\ell_1$-norm of the filter DFT. In this paper, we discuss a new family of adaptive estimates which rely upon minimizing the $\ell_2$-norm of the estimation residual. We show that such estimators possess better statistical properties than those based on $\ell_\infty$-fit; in particular, we prove oracle inequalities for their $\ell_2$-loss and improved bounds for $\ell_2$- and pointwise losses. The oracle inequalities rely on the "approximate shift-invariance" assumption stating that the signal to be recovered is close to an (unknown) shift-invariant subspace. We also study the relationship of the approximate shift-invariance assumption with the "signal simplicity" assumption introduced in Juditsky and Nemirovski (2009) and discuss the application of the proposed approach to harmonic oscillations denoising.