Measures of scalability

TL;DR

This paper introduces three new quantitative measures to evaluate how close a finite-dimensional frame is to being scalable, providing theoretical bounds and conditions for scalability.

Contribution

It proposes three novel measures of scalability, proves their equivalence, and derives new conditions and probabilistic bounds for frame scalability.

Findings

Three measures of scalability are introduced and shown to be equivalent.

Bounds on the probability of random frames being scalable are established.

New necessary and sufficient conditions for frame scalability are derived.

Abstract

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.