Overview of recent advances in numerical tensor algebra

We present a survey of some recent developments for decompositions of multi-way arrays or tensors, with special emphasis on results relevant for applications and modeling in signal processing. A central problem is how to find low-rank approximations of tensors, and we describe some new results, including numerical methods, algorithms and theory, for the higher order singular value decomposition (HOSVD) and the parallel factors expansion or canonical decomposition (CP expansion).