Published on Mon Jun 11 2018

### Adaptive Denoising of Signals with Local Shift-Invariant Structure

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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. We show that such estimators possess better statistical properties than those based on fit.

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###### 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 -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 -norm of the Discrete Fourier Transform (DFT) of the estimation residual is minimized under constraint on the -norm of the filter DFT. In this paper, we discuss a new family of adaptive estimates which rely upon minimizing the -norm of the estimation residual. We show that such estimators possess better statistical properties than those based on -fit; in particular, we prove oracle inequalities for their -loss and improved bounds for - 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.