Denoising and Filtering Under the Probability of Excess Loss Criterion

Subclasses of finite alphabet denoising and filtering (causal denoising) schemes are compared. Performance is measured by the normalized cumulative loss (a.k.a. distortion), as measured by a single-letter loss function. We aim to minimize the probability that the normalized cumulative loss exceeds a given threshold. We call this quantity the probability of excess loss. Specifically, we consider a scheme to be optimal if it attains the maximal exponential decay rate of the probability of excess loss. This provides another way of comparing schemes that complements and contrasts previous work which considered the expected value of the normalized cumulative loss. In particular, the question of whether the optimal denoiser is symbol-by-symbol for an independent and identically distributed (i.i.d.) source and a discrete memoryless channel (DMC) is investigated. For Hamming loss, the optimal denoiser is proven to be symbol-by-symbol. Perhaps somewhat counterintuitively, for a general single letter loss function, the optimal scheme need not be symbol-by-symbol. The optimal denoiser requires unbounded delay and unbounded look-ahead while symbol-by-symbol schemes mandate zero delay and look-ahead. It is natural to wonder about the effect of limited delay and limited look-ahead. Consequently, finite sliding-window denoisers and finite block denoisers are defined. They are shown to perform no better than symbol-by-symbol denoisers. Finally, the effect of causality is investigated. While it is difficult to characterize the performance of filters with unbounded memory explicitly, it is shown that finite memory filters perform no better than symbol-by-symbol filters