Using frames in statistical signal recovering

Overcomplete representations such as wavelets and windowed Fourier expansions have become mainstays of modern statistical data analysis. Here we derive expressions for the mean quadratic risk of shrinkage estimators in the context of general finite frames, which include any fullrank linear expansion of vector data in a finite-dimensional setting. We provide several new results and practical estimation procedures that take into account the geometric correlation structure of frame elements. These results motivate aggregation estimators and block thresholding procedures, and reinforce that the correlations induced by frame structure should be explicitly treated to yield improvements in estimation. A simulation study confirms these improvements.