Full Spark Frames

TL;DR

This paper introduces new methods for constructing full spark frames using the discrete Fourier transform, proves their abundance among Parseval frames, and shows that testing for full spark property is computationally hard.

Contribution

It provides novel deterministic constructions of full spark frames, proves their density among Parseval frames, and establishes the computational difficulty of testing the full spark property.

Findings

Full spark frames can be constructed using the discrete Fourier transform.

Full spark Parseval frames are dense in all Parseval frames.

Testing whether a matrix is full spark is NP-hard.

Abstract

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M-dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.