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On concentration for denoiser-loss estimators

We study the concentration of denoiser loss estimators, with application to the selection of denoiser parameters for a given observed sequence (in particular, the window size k of the DUDE algorithm [1]) via minimization of the estimated loss. We show that for a loss estimator proposed earlier [2], it is not possible to derive strong concentration results for certain pathological input sequences. By modifying the estimator slightly we obtain a loss estimator for which the DUDE's estimated loss strongly concentrates around the true loss provided kM2k = o(n), where M is the size of the alphabet and n the sequence length. We also show that for certain channels, it is possible to estimate the best k using a combination of the two loss estimators. Moreover, for non-pathological sequences and k = o(n¼), we derive concentration results for the original loss estimator and all channels.