Frame Scalings: A Condition Number Approach

TL;DR

This paper investigates the problem of optimally scaling frames to improve their condition number, formulating it as a convex optimization problem involving the operator norm, with implications for signal processing applications.

Contribution

It introduces a novel convex optimization approach for finding the best conditioned frame via scaling, extending previous work focused on the Frobenius norm.

Findings

The problem is equivalent to a convex optimization involving the operator norm.

Analysis of Frobenius norm case related to frame operator condition number.

Convexity properties of optimal scalings are established.

Abstract

Scaling frame vectors is a simple and noninvasive way to construct tight frames. However, not all frames can be modifed to tight frames in this fashion, so in this case we explore the problem of finding the best conditioned frame by scaling, which is crucial for applications like signal processing. We conclude that this problem is equivalent to solving a convex optimization problem involving the operator norm, which is unconventional since this problem was only studied in the perspective of Frobenious norm before. We also further study the Frobenious norm case in relation to the condition number of the frame operator, and the convexity of optimal scalings.