Partition mixture of 1D wavelets for multi-dimensional data
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Traditional statistical wavelet analysis that carries out modeling and inference based on wavelet coefficients under a given, predetermined wavelet transform can quickly lose efficiency in multivariate problems, because such wavelet transforms, which are typically symmetric with respect to the dimensions, cannot adaptively exploit the energy distribution in a problem-specific manner. We introduce a principled probabilistic framework for incorporating such adaptivity---by (i) representing multivariate functions using one-dimensional (1D) wavelet transforms applied to a permuted version of the original function, and (ii) placing a prior on the corresponding permutation, thereby forming a mixture of permuted 1D wavelet transforms. Such a representation can achieve substantially better energy concentration in the wavelet coefficients. In particular, when combined with the Haar basis, we show that exact Bayesian inference under the model can be achieved analytically through a recursive message passing algorithm with a computational complexity that scales linearly with sample size. In addition, we propose a sequential Monte Carlo (SMC) inference algorithm for other wavelet bases using the exact Haar solution as the proposal. We demonstrate that with this framework even simple 1D Haar wavelets can achieve excellent performance in both 2D and 3D image reconstruction via numerical experiments, outperforming state-of-the-art multidimensional wavelet-based methods especially in low signal-to-noise ratio settings, at a fraction of the computational cost.
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