Superfast Wavelet Transform Using Quantics-TT Approximation. I. Application to Haar Wavelets

We propose a superfast discrete Haar wavelet transform (SFHWT) as well as its inverse, using the low-rank Quantics-TT (QTT) representation for the Haar transform matrices and input-output vectors. Though the Haar matrix itself does not have a low QTT rank approximation, we show that factor matrices used at each step of the traditional multilevel Haar wavelet transform algorithm have explicit QTT representations of low rank. The SFHWT applies to a vector representing a signal sampled on a uniform grid of size N = 2 d ${N=2^d}$ . We develop two algorithms which roughly require square logarithmic time complexity O ( log 2 N ) ${O(\log ^2 N)}$ with respect to the grid size, hence outperforming the traditional fast Haar wavelet transform (FHWT) of linear complexity O(N). Our approach also applies to the FHWT inverse as well as to the multidimensional wavelet transform. Numerical experiments demonstrate that the SFHWT algorithm is robust in keeping low rank of the resulting output vector and it outperforms the traditional FHWT for grid sizes larger than a certain value depending on the spacial dimension.