Tensor decompositions of positive definite tensors

Tensors are a natural generalization of matrices to higher order settings. These multi-indexed arrays are of great applied and theoretical interest due to the many surprising properties they exhibit. For example, the rank decomposition of a low rank tensor is (essentially) unique under very light assumptions — enabling recovery of underlying information from signal tensors of interest. Another surprising fact is that the set of tensors of some fixed rank is typically not closed.

This talk provides an introduction to tensor decompositions and their use in applications, discusses some of the interesting phenomena exhibited by tensors, and presents recent results on the closedness of the set of low rank tensors: the set of low rank “positive definite” tensors is relatively closed as a subset of the set of positive definite tensors. Based on joint work with Lieven De Lathauwer.