Linear Algebra Seminar

Speaker: Luke Oeding

Title: Homotopy techniques for tensor decomposition and perfect identifiability

Abstract: Given a tensor (or hyper-matrix), we would like to express it in the simplest possible way as the sum of the smallest number of decomposable (or rank-1) tensors. While there are many algorithms that attempt to accomplish this task, it is known to be a very difficult problem.  Moreover, such a decomposition may not be unique.  When a generic tensor of a given format has a unique decomposition, we say that tensors of that format are "generically identifiable."

We propose a new method to find tensor decompositions via homotopy continuation. This technique allows us to find all decompositions of a given tensor (at least for relatively small tensors).  Our experiments yielded a surprise - we found two new tensor formats, (3,4,5) and (2,2,2,3), where the generic tensor has a unique decomposition. Using techniques from algebraic geometry, we prove that these cases are indeed "generically identifiable".

This is joint work with J. Hauenstein, G. Ottaviani and A. Sommese.