Optimization and the Topology of Spaces of Parseval Frames
Anthony Caine, Tom Needham, Clayton Shonkwiler
TL;DR
This work addresses the problem of constructing Parseval frames with prescribed column norms and understanding the topology of the associated frame spaces. It introduces the total frame energy $E_{\vec{r}}$ whose global minima are the spaces $\operatorname{PF}_d(\vec{r})$ and proves a Benedetto–Fickus–type result: for rational $\vec{r}$, all local minima are global, enabling gradient flow to converge to a member of $\operatorname{PF}_d(\vec{r})$. Using this, the authors realize frame spaces as deformation retracts of semistable ambient spaces and derive vanishing results for homotopy groups, yielding new path-connectedness statements for real and complex Parseval frame spaces. The approach blends optimization via momentum-map–inspired energies with Geometric Invariant Theory, providing a framework that links frame theory, topology, and symplectic methods, with potential implications for both theory and signal-processing applications.
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.