Seminar Details
2025-12-16 (14:00) : Geometry of low-rank tensors
At EULER (room A.002)
Organized by Mathematical Engineering
Speaker :
Simon Jacobsson (KU Leuven)
Abstract :
Morally, a manifold is a set where notions from calculus are well-defined. For example, the set of n-by-n orthogonal matrices and the set of n-by-n symmetric positive definite matrices are manifolds. Knowing that a set of matrices or tensors is a manifold allows us to use a host of calculus tools to do numerical analysis on that set. For example, many constrained optimization problems can be formulated as unconstrained manifold optimization problems. Algorithms for these can then make use of manifold gradients and analogues of straight lines called geodesics.
We consider the set of fixed-rank tensors. When the rank is sufficiently low, then (almost) any tensor in this set is related to (almost) any other tensor via a change of basis. We explain how this relation induces a manifold structure, and show how relevant quantities like gradients and geodesics can be computed efficiently. More precisely, we identify the set as a quotient of Lie groups. We also discuss how the manifold perspective can be used to integrate tensor differential equations.