Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry, and numerical optimization to the group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths and polynomial paths, and discuss open problems concerning the condition number of the recovery problem. Based on joint work with Max Pfeffer and Bernd Sturmfels.
- Numerical Analysis Group Internal Seminar