Denoising as well as the best of any two denoisers

TL;DR

This paper investigates whether a new denoiser can be constructed to outperform two given denoisers asymptotically, revealing conditions under which this is possible or impossible, and providing solutions for specific channels.

Contribution

It introduces a method to combine two denoisers to achieve near-optimal performance, identifies limitations of loss estimation, and offers a solution for the binary symmetric channel.

Findings

Loss estimation works under certain restrictions.

Counter-example shows limitations of the approach.

Randomization enables optimal denoising for binary symmetric channel.

Abstract

Given two arbitrary sequences of denoisers for block lengths tending to infinity we ask if it is possible to construct a third sequence of denoisers with an asymptotically vanishing (in block length) excess expected loss relative to the best expected loss of the two given denoisers for all clean channel input sequences. As in the setting of DUDE [1], which solves this problem when the given denoisers are sliding block denoisers, the construction is allowed to depend on the two given denoisers and the channel transition probabilities. We show that under certain restrictions on the two given denoisers the problem can be solved using a straightforward application of a known loss estimation paradigm. We then show by way of a counter-example that the loss estimation approach fails in the general case. Finally, we show that for the binary symmetric channel, combining the loss estimation with a randomization step leads to a solution to the stated problem under no restrictions on the given denoisers.