Breaking the curse of dimensionality
In our new publication, “Breaking the curse of dimensionality, or how to use SVD in many dimensions” we present a simple recursive decomposition for multidimensional functions, operators, vectors and matrices. We prove that in the case when the canonical decomposition exists our new Tree-Tucker decomposition also exists with the same (and often fewer) number of parameters. However, unlike the canonical decomposition, the Tree-Tucker format is stable and robustly computable by exploiting the SVD decomposition (or any rank-revealing decomposition). We show how to perform a simple operation (multidimensional convolution) in such format and provide numerical experiments for the dimensions up to d=200 on a notebook in several minutes.