New results on sparse representations in unions of orthonormal bases
Tao Zhang, Gennian Ge
TL;DR
This work addresses constructing dictionaries that maximize the spark for a dictionary formed by a union of orthonormal bases. It introduces a finite-field–based construction with $q=2^r$, $n=q^{2t}$, and $d=q+1$ bases, producing $\eta(D)=q^t+q^{t-1}$ and $\mu(D)=\frac{1}{q^t}$, thus attaining the Gribonval–Nielsen bound. The main technique combines trace functions $\mathrm{Tr}_{m}^{n}$ and carefully designed basis families $B_a$ and $B_{q+1}$ to ensure orthogonality within bases and controlled cross-coherence between bases. This extends Shen et al. (2022) by providing an explicit dictionary achieving the bound for $(n,d)=(q^{2t},q+1)$ with $q=2^r$. The results have implications for sparse representation guarantees in compressed sensing and related signal-processing applications where large spark and low coherence are advantageous.
Abstract
The problem of sparse representation has significant applications in signal processing. The spark of a dictionary plays a crucial role in the study of sparse representation. Donoho and Elad initially explored the spark, and they provided a general lower bound. When the dictionary is a union of several orthonormal bases, Gribonval and Nielsen presented an improved lower bound for spark. In this paper, we introduce a new construction of dictionary, achieving the spark bound given by Gribonval and Nielsen. Our result extends Shen et al.' s findings [IEEE Trans. Inform. Theory, vol. 68, pp. 4230--4243, 2022].