Adaptive thresholding of sequences with locally variable strength

This paper addresses, via thresholding, the estimation of a possibly sparse signal observed subject to Gaussian noise. Conceptually, the optimal threshold for such problems depends upon the strength of the underlying signal. We propose two new methods that aim to adapt to potential local variation in this signal strength and select a variable threshold accordingly. Our methods are based upon an empirical Bayes approach with a smoothly variable mixing weight chosen via either spline or kernel based marginal maximum likelihood regression. We demonstrate the excellent performance of our methods in both one and two-dimensional estimation when compared to various alternative techniques. In addition, we consider the application to wavelet denoising where reconstruction quality is significantly improved with local adaptivity.