Wavelets and the theory of non-parametric function estimation

Non-parametric function estimation aims to estimate or recover or denoise a function of interest, perhaps a signal, spectrum or image, that is observe d in noise and possibly indirectly after some transformation, as in deconvo lution. 'Non-parametric' signifies that no a priori: limit is placed on the number of unknown parameters used to model the signal. Such theories of es timation are necessarily quite different from traditional statistical model s with a small number of parameters specified in advance. Before wavelets, the theory was dominated by linear estimators, and the exp loitation of assumed smoothness in the unknown function to describe optimal methods. Wavelets provide a set of tools that make it natural to assert, i n plausible theoretical models, that the sparsity of representation is a mo re basic notion than smoothness, and that nonlinear thresholding can be a p owerful competitor to traditional linear methods. We survey some of this st ory, showing how sparsity emerges from an optimality analysis via the game- theoretic notion of a least-favourable distribution.