A quiver invariant theoretic approach to Radial Isotropy and Paulsen's Problem for matrix frames

TL;DR

This paper applies quiver invariant theory to study matrix frames, generalizing Radial Isotropy and addressing Paulsen's problem by establishing bounds on frame perturbations.

Contribution

It introduces a quiver invariant theoretic framework for matrix frames, extending Radial Isotropy Theorem and providing bounds for the Paulsen problem.

Findings

Generalized Radial Isotropy Theorem for matrix frames

Established bounds for nearly equal-norm Parseval frames

Provided a quiver invariant approach to Paulsen's problem

Abstract

In this paper, we view matrix frames as representations of quivers and study them within the general framework of quiver invariant theory. We are thus led to consider the large class of semi-stable matrix frames. Within this class, we are particularly interested in radial isotropic and Parseval matrix frames. Using methods from quiver invariant theory [CD19], we first prove a far reaching generalization of Barthe's Radial Isotropy Theorem [Bar98] to matrix frames (see Theorems 1(3) and 29). With this tool at our disposal, we provide a quiver invariant theoretic approach to the Paulsen problem for matrix frames. We show in Theorem 2 that for any given $\varepsilon$-nearly equal-norm Parseval frame $\mathcal{F}$ of $n$ matrices with $d$ rows there exists an equal-norm Parseval frame $\mathcal{W}$ of $n$ matrices with $d$ rows such that $\mathsf{dist}^2(\mathcal{F},\mathcal{W})\leq 46 \epsilon d^2$.