The Degree of Tensor Train Varieties via Random Algebraic Geometry

Originating in the study of quantum many-body systems, tensor trains have found a wide range of applications, including machine learning, wave-function compression, and fluid dynamics. They belong to a broader family of tensor decompositions, alongside tree tensor networks and Tucker decompositions. As such, tensor trains naturally give rise to parametrized algebraic varieties. While the dimensions of these varieties are known, their degrees are usually much harder to determine. In this talk, I will present an algorithmic procedure for computing the degree of tensor train varieties. The approach draws on ideas from random algebraic geometry and integral geometry, allowing one to compute the degree of a projective variety through its volume.