On Structural Decompositions of Finite Frames

TL;DR

This paper explores the combinatorial structure of finite frames in Hilbert spaces, introducing posets to analyze their decompositions into tight or scalable subsets, and investigates the inverse problem of realizing specific poset structures.

Contribution

It defines factor and scalability posets for frames, establishes conditions for their properties, and studies the inverse problem of constructing frames with prescribed poset structures.

Findings

Necessary and sufficient conditions for factor posets in $H_2$

Preservation of factor poset structure under orthogonal projections

Bounds on the size and enumeration of factor posets

Abstract

A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be scalable if there exist nonnegative scalars $\{c_i\}_{i\in I}$ such that $\{c_if_i\}_{i\in I}$ is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame $\{f_i\}_{i\in I}$ to be a collection of subsets of $I$ ordered by inclusion so that nonempty $J\subseteq I$ is in the factor poset if and only if $\{f_j\}_{j\in J}$ is a tight frame for $H_n$. A similar definition is given for the scalability poset of a frame. We prove conditions which factor posets satisfy and use these to study the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset $P$. We determine a necessary condition for solving the inverse factor poset problem in $H_n$ which is also sufficient for $H_2$. We describe how factor poset structure of frames is preserved under orthogonal projections. We also consider the enumeration of the number of possible factor posets and bounds on the size of factors posets. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset.