Algorithms for Discrete Denoising Under Channel Uncertainty

TL;DR

This paper develops practical algorithms for discrete signal denoising when the noise channel is uncertain, focusing on scalar-parameterized channels, and demonstrates their efficiency and effectiveness through theoretical analysis and empirical results.

Contribution

It introduces efficient, convex-optimization-based denoising algorithms for scalar-parameterized channels under uncertainty, extending previous asymptotic optimality results to practical implementation.

Findings

Convex optimization approach for symmetric channels.

Efficient estimator for feasible channel subset.

Modified discrete universal denoiser performs well in practice.

Abstract

The goal of a denoising algorithm is to reconstruct a signal from its noise-corrupted observations. Perfect reconstruction is seldom possible and performance is measured under a given fidelity criterion. In a recent work, the authors addressed the problem of denoising unknown discrete signals corrupted by a discrete memoryless channel when the channel, rather than being completely known, is only known to lie in some uncertainty set of possible channels. A sequence of denoisers was derived for this case and shown to be asymptotically optimal with respect to a worst-case criterion argued most relevant to this setting. In the present work we address the implementation and complexity of this denoiser for channels parametrized by a scalar, establishing its practicality. We show that for symmetric channels, the problem can be mapped into a convex optimization problem, which can be solved efficiently. We also present empirical results suggesting the potential of these schemes to do well in practice. A key component of our schemes is an estimator of the subset of channels in the uncertainty set that are feasible in the sense of being able to give rise to the noise-corrupted signal statistics for some channel input distribution. We establish the efficiency of this estimator, both algorithmically and experimentally. We also present a modification of the recently developed discrete universal denoiser (DUDE) that assumes a channel based on the said estimator, and show that, in practice, the resulting scheme performs well. For concreteness, we focus on the binary alphabet case and binary symmetric channels, but also discuss the extensions of the algorithms to general finite alphabets and to general channels parameterized by a scalar.