Title: WARP: Wavelets with adaptive recursive partitioning for multi-dimensional data
Authors: Meng Li - Duke University (United States)
Li Ma - Duke University (United States) [presenting]
Abstract: Traditional statistical wavelet analysis carries out modeling and inference under a given, predetermined wavelet transform. This approach can quickly lose efficiency for multi-dimensional data (e.g., observations measured on a multi-dimensional grid), because a predetermined transform does not exploit the structure of the underlying function in a problem-specific manner. We overcome this challenge by making the wavelet transform adaptive to the structure of the data. By exploiting a connection between permutations on the index space of multi-dimensional functions and recursive partitions on that space, we show that the desired adaptivity in the wavelet transform can be achieved through Bayesian hierarchical modeling on the space of such recursive partitions. When one applies this framework to Haar wavelets, exact Bayesian inference under the model can be achieved analytically through recursive message passing with an efficient computational complexity linear in the sample size. We also provide recipes for incorporating block shrinkage into the framework as well as for applying it to other wavelet bases. We demonstrate via numerical experiments that with the enhancement under this framework even simple Haar wavelets can achieve excellent performance in 2D and 3D image reconstruction. We investigate the source of the gain by quantitatively comparing the efficacy of energy concentration under our adaptive wavelet transforms to that of classical fixed wavelet transforms.