Codes and Expansions (CodEx) Seminar

Clay Shonkwiler (Colorado State University)
Optimization and Special Matrices

We often identify matrices of interest by prescribing features: having a given spectrum, being self-adjoint or bistochastic, etc. Once we have specified some such collection of matrices, questions immediately arise: Is it nonempty? How could I find an element in this collection? Is it possible to interpolate between two such matrices?

In this talk I will describe an optimization-based approach to such questions for a number of different classes of special matrices, including equal-norm Parseval frames, normal matrices, and (adjacency matrices of) balanced graphs. In each case, the space of special matrices can be realized as the deformation retract of a larger, simpler space of matrices via gradient descent of a natural loss function. This gives both a procedure for constructing special matrices (do gradient descent from any random starting point) and substantial information about the topology of the space of special matrices. In particular, this proves various generalizations and strengthenings of Larson's Frame Homotopy Conjecture. This is joint work with Tom Needham and Anthony Caine.