Denoising as well as the best of any two denoisers

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.