Scalable Frames

TL;DR

This paper investigates the concept of scalable frames, which are frames that can be transformed into tight frames through diagonal rescaling, providing characterizations and geometric insights applicable in finite and infinite dimensions.

Contribution

It introduces the notion of scalable frames, offers multiple characterizations including in infinite dimensions, and provides a geometric interpretation of scalability.

Findings

Characterization of scalable frames in finite and infinite dimensions

Existence conditions for diagonal rescaling to achieve tight frames

Geometric interpretation of scalability via conical surfaces

Abstract

Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.