Normalizing Parseval frames by gradient descent

Abstract

Equinorm Parseval Frames (ENPFs) are collections of equal-length vectors that form Parseval frames, meaning they are spanning sets that satisfy a version of the Parseval identity. As such, they have many of the desirable features of orthonormal bases for signal processing and data representation, but provide advantages over orthonormal bases in settings where redundancy is important to provide robustness to data loss. We give three methods for normalizing Parseval frames: that is, flowing a generic Parseval frame to an ENPF. This complements prior work showing that equal-norm frames could be "Parsevalized" and potentially provides new avenues for attacking the Paulsen problem, which seeks sharp upper bounds on the distance to the space of ENPFs in terms of norm and spectral data. This work is based on ideas from symplectic geometry and geometric invariant theory.