Adaptive Denoising of Signals with Local Shift-Invariant Structure

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 the convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski [20] 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, under the assumption of approximate shift-invariance we prove oracle inequalities for their $$\ell _2$$ -loss and improved bounds for $$\ell _2$$ - and pointwise losses. We also study the relationship of the approximate shift-invariance assumption with the signal simplicity introduced in [20], and discuss the application of the proposed approach to harmonic oscillation denoising.