Sparsity and spectral properties of dual frames

TL;DR

This paper investigates the sparsity and spectral characteristics of dual frames in finite-dimensional spaces, providing bounds, exact sparsity formulas, and spectral pattern characterizations, including explicit constructions for certain duals.

Contribution

It introduces bounds on dual sparsity, an exact formula for the sparsest dual using generalized spark, and characterizes spectral patterns of dual frames with explicit constructions.

Findings

Any finite frame has a dual with at most n^2 non-zero entries.

Most frames do not admit sparser duals than the constructed bounds.

Explicit duals with prescribed spectra can be constructed, especially tight duals.

Abstract

We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than $n^2$ non-vanishing entries, where $n$ denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion of spark. We then study the spectral properties of dual frames in terms of singular values of the synthesis operator. We provide a complete characterization for which spectral patterns of dual frames are possible for a fixed frame. For many cases, we provide simple explicit constructions for dual frames with a given spectrum, in particular, if the constraint on the dual is that it be tight.