Optimization methods for frame conditioning and application to graph Laplacian scaling

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.

18 pages, 3 figures. To appear in Novel methods in harmonic analysis with applications to numerical analysis and data processing", Lecture Notes ANHA Series, I. Pesenson and all Eds., Birkhauser (2017)