On Optimal Frame Conditioners

TL;DR

This paper explores the problem of making frames tight through rescaling, reformulating it as a convex optimization problem and demonstrating its effectiveness with numerical experiments.

Contribution

It introduces a convex optimization approach to determine when a frame can be scaled to become tight, providing new methods and numerical results.

Findings

Frames can be scaled to tight frames using convex optimization.

The proposed methods successfully identify scalable frames in experiments.

Numerical results demonstrate the effectiveness of the approach.

Abstract

A (unit norm) frame is scalable if its vectors can be rescaled so as to result into a tight frame. Tight frames can be considered optimally conditioned because the condition number of their frame operators is unity. In this paper we reformulate the scalability problem as a convex optimization question. In particular, we present examples of various formulations of the problem along with numerical results obtained by using our methods on randomly generated frames.