Minimal scalings and structural properties of scalable frames

TL;DR

This paper investigates the properties of scalable frames in real Euclidean spaces, focusing on minimal scalings, their structural dependencies, and the uniqueness of orthogonal partitions, to better understand their geometric and algebraic characteristics.

Contribution

It provides bounds on the number of minimal scalings, characterizes affine dependence among them, and establishes the uniqueness of orthogonal partitioning in scaled frames.

Findings

Number of minimal scalings can be estimated.

Minimal scalings are affinely dependent under certain conditions.

All strict scalings share the same structural properties.

Abstract

For a unit-norm frame $F = \{f_i\}_{i=1}^k$ in $\R^n$, a scaling is a vector $c=(c(1),\dots,c(k))\in \R_{\geq 0}^k$ such that $\{\sqrt{c(i)}f_i\}_{i =1}^k$ is a Parseval frame in $\R^n$. If such a scaling exists, $F$ is said to be scalable. A scaling $c$ is a minimal scaling if $\{f_i : c(i)>0\}$ has no proper scalable subframe. It is known that the set of all scalings of $F$ is a convex polytope whose vertices correspond to minimal scalings. In this paper, we provide an estimation of the number of minimal scalings of a scalable frame and a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings $c=(c(1),\dots,c(k))\in \R_{> 0}^k$ of $F$ have the same structural property. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame.