Optimization and the Topology of Spaces of Parseval Frames

TL;DR

This paper studies the structure of spaces of finite-dimensional Parseval frames with prescribed norms, showing gradient descent convergence and topological properties like path-connectedness, extending existing theorems in frame theory.

Contribution

It introduces a nonconvex function measuring deviations from Parseval and norm constraints, proves the absence of spurious local minima, and applies this to analyze the topology of frame spaces.

Findings

Gradient descent converges to equal norm Parseval frames from dense initial conditions.

Spaces of Parseval frames with prescribed norms are deformation retracts of simpler spaces.

New conditions for vanishing homotopy groups and path-connectedness of frame spaces.

Abstract

A Parseval frame is a spanning set for a Hilbert space which satisfies the Parseval identity: a vector can be expressed as a linear combination of the frame whose coefficients are inner products with the frame vectors. There is considerable interest within the signal processing community in the structural properties of the space of finite-dimensional Parseval frames whose vectors all have the same norm, or which satisfy more general prescribed norm constraints. In this paper, we introduce a function on the space of arbitrary spanning sets that jointly measures the failure of a spanning set to satisfy both the Parseval identity and given norm constraints. We show that, despite its nonconvexity, this function has no spurious local minimizers, thereby extending the Benedetto--Fickus theorem to this non-compact setting. In particular, this shows that gradient descent converges to an equal norm Parseval frame when initialized within a dense open set in the associated matrix space. We then apply this result to study the topology of frame spaces. Using our Benedetto--Fickus-type result, we realize spaces of Parseval frames with prescribed norms as deformation retracts of simpler spaces, leading to explicit conditions which guarantee the vanishing of their homotopy groups. These conditions yield new path-connectedness results for spaces of real Parseval frames, generalizing the Frame Homotopy Theorem, which has seen significant interest in recent years.