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Lower Limits of Discrete Universal Denoising

Viswanathan, Krishnamurthy; Ordentlich, Erik

HPL-2006-71

Keyword(s): denoising; universal algorithms; individual sequences; discrete memoryless channels

Abstract: In the spirit of results on universal compression, we compare the performance of universal denoisers on discrete memoryless channels to that of the best performance obtained by a k-th order omniscient denoiser, namely one that is tuned to the transmitted noiseless sequence. We show that the additional loss incurred in the worst case by any universal denoiser on a length-n sequence grows at least like , where c is a constant depending on the channel parameters and the loss function. This shows that for fixed k the additional loss incurred by the Discrete Universal Denoiser (DUDE) derived by Weissman et al is no larger than a constant multiplicative factor. Furthermore we compare universal denoisers to denoisers that are aware of the distribution of the transmitted noiseless sequence. We show that, even for this weaker target loss, for any universal denoiser, there exists some i.i.d. noiseless distribution whose optimum expected loss is lower than that incurred by the universal denoiser by .

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