Optimization methods for frame conditioning and application to graph Laplacian scaling

TL;DR

This paper investigates optimization techniques for determining frame scalability and applies these methods to reweight graphs to improve the Laplacian's condition number, enhancing spectral properties.

Contribution

It introduces new algorithms for frame scalability and extends these methods to optimize graph Laplacian reweighting for better spectral conditioning.

Findings

Algorithms successfully determine frame scalability

Reweighting graphs reduces Laplacian condition number

Improved spectral properties of graphs through optimization

Abstract

A frame is scalable if each of its vectors can be rescaled in such a way that the resulting set becomes a Parseval frame. In this paper, we consider four different optimization problems for determining if a frame is scalable. We offer some algorithms to solve these problems. We then apply and extend our methods to the problem of reweighing (finite) graph so as to minimize the condition number of the resulting Laplacian.