Denoising as well as the best of any two denoisers

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.Comment: 19 pages. Appeared, in part, in Proceedings of 2013 IEEE Intl. Symp. on Info. Theory. This version has full proofs (e.g., of Proposition 2

A Denoising Loss Bound for Neural Network based Universal Discrete Denoisers

We obtain a denoising loss bound of the recently proposed neural network based universal discrete denoiser, Neural DUDE, which can adaptively learn its parameters solely from the noise-corrupted data, by minimizing the \emph{empirical estimated loss}. The resulting bound resembles the generalization error bound of the standard empirical risk minimizers (ERM) in supervised learning, and we show that the well-known bias-variance tradeoff also exists in our loss bound. The key tool we develop is the concentration of the unbiased estimated loss on the true denoising loss, which is shown to hold \emph{uniformly} for \emph{all} bounded network parameters and \emph{all} underlying clean sequences. For proving our main results, we make a novel application of the tools from the statistical learning theory. Finally, we show that the hyperparameters of Neural DUDE can be chosen from a small validation set to significantly improve the denoising performance, as predicted by the theoretical result of this paper.Comment: submitted to ICML 201