A note on scalable frames

TL;DR

This paper investigates the problem of determining the scalability of frames, showing that most are either non-scalable or uniquely scalable, and provides methods to find and analyze all possible scalings.

Contribution

It introduces a simple linear system approach for checking scalability and characterizes the set of all scalings as a convex polytope with minimal scalings as vertices.

Findings

Most frames are either not scalable or have a unique scaling.

The set of all scalings forms a convex polytope.

Vertices of the polytope correspond to minimal scalings.

Abstract

We study the problem of determining whether a given frame is scalable, and when it is, understanding the set of all possible scalings. We show that for most frames this is a relatively simple task in that the frame is either not scalable or is scalable in a unique way, and to find this scaling we just have to solve a linear system. We also provide some insight into the set of all scalings when there is not a unique scaling. In particular, we show that this set is a convex polytope whose vertices correspond to minimal scalings.