The tensor structure of a class of adaptive algebraic wavelet transforms

We revisit the wavelet tensor train (WTT) transform, an algebraic orthonormal wavelettype transform based on the successive separation of variables in multidimensional arrays, or tensors, underlying the tensor train (TT) representation of such arrays. The TT decomposition was proposed for representing and manipulating data in terms of relatively few parameters chosen adaptively, and the corresponding low-rank approximation procedure may be seen as the construction of orthonormal bases in certain spaces associated to the data. Using these bases, which are extracted adaptively from a reference vector, to represent other vectors is the idea underlying the WTT transform. When the TT decomposition is coupled with the quantization of “physical” dimensions, it seeks to separate not only the “physical” indices, but all “virtual” indices corresponding to the virtual levels, or scales, of those. This approach and the related construction of the WTT transform are closely connected to hierarchic multiscale bases and to the framework of multiresolution analysis. In the present paper we analyze the tensor structure of the WTT transform. First, we derive an explicit TT decomposition of its matrix in terms of that of the reference vector. In particular, we establish a relation between the ranks of the two representations, which govern the numbers of parameters involved. Also, for a vector given in the TT format we construct an explicit TT representation of its WTT image and bound the TT ranks of the representation. Finally, we analyze the sparsity of the WTT basis functions at every level and show the exponential reduction of their supports, from the coarsest level to the finest level, with respect to the level number.