Bivariate hard thresholding in wavelet function estimation

We propose a generic bivariate hard thresholding estimator of the discrete wavelet coefficients of a function contaminated with i.i.d. Gaussian noise. We demonstrate its good risk properties in a motivating example, and derive upper bounds for its mean-square error. Motivated by the clustering of large wavelet coefficients in real-life signals, we propose two wavelet denoising algorithms, both of which use specific instances of our bivariate estimator. The BABTE algorithm uses basis averaging, and the BITUP algorithm uses the coupling of ``parents" and ``children" in the wavelet coefficient tree. We prove the near-optimality of both algorithms over the usual range of Besov spaces, and demonstrate their excellent finite-sample performance. Finally, we propose a robust and effective technique for choosing the parameters of BITUP in a data-driven way.