Tensor Decomposition

Lecture 1: Tensors and Classical Algebraic Geometry
Lecture 2: Algorithms for Tensor Decomposition and Identifiability

This series of talks was part of the Algebraic Geometry Boot Camp. Videos for each talk are available through the links above.

Speaker: Luke Oeding, Auburn University

Lecture 1: I will introduce tensors and tensor rank from an algebraic perspective. I will introduce multilinear rank and tensor rank, and I will discuss the related classical algebraic varieties — subspace varieties and secant varieties. I will give basic tools for computing dimensions, Terracini's lemma and the notion of the abstract secant variety. This will lead to the notion of generic rank. I will briefly disucss implicit equations, which lead to rank tests.

Lecture 2: I will focus on one special case of tensor decomposition — symmetric tensors and Waring decomposition. I will start by discussing the naive approach, then I will discuss Sylvester's algorithm for binary forms. As a bonus I will show how Sylvester's algorithm for symmetric tensor decomposition also gives a method find the roots of a cubic polynomial in one variable.

I will discuss what to expect regarding generic rank and uniqueness of tensor decomposition. With the remaining time I will discuss the recently defined notion of an eigenvector of a (symmetric) tensor (Lim05, Qi05), which leads to a new method (developed with Ottaviani) for exact Waring decomposition.