# Codes and Expansions (CodEx) Seminar

## Elina Robeva (University of British Columbia)Orthogonal and Incoherent Tensor decomposition

Tensor decomposition has many applications. However, it is often a hard problem. In this talk we will discuss two families of tensors with special structure which retain some of the properties of matrices that general tensors don't. A symmetric orthogonally decomposable tensor can be written as a linear combination of tensor powers of n orthonormal vectors. As opposed to general tensors, such tensors can be decomposed efficiently. We study the spectral properties of such tensors and give a formula for all of their eigenvectors. We also give polynomial equations defining the set of all orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors. To extend the definition to a larger set of tensors, we define equiangular tight-frame decomposable tensors and study their properties. We show that if the tight frame is a Mercedez-Benz frame, then, the tensor power method finds a tensor decomposition. We conclude with some open questions and future research directions.