We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of sparks are generally combinatorial. For unions of several orthonormal bases, two lower bounds on the spark via the mutual coherence were established in previous work. We constructively prove that both of them are tight. Our main results give positive answers to Gribonval and Nielsen’s open problem on sparse representations in unions of orthonormal bases. Constructive proofs rely on a family of mutually unbiased bases which first appears in quantum information theory.

One-bit compressive sensing theory shows that the sparse signals can be almost exactly reconstructed from a small number of one-bit quantized linear measurements. This paper presents the convergence analysis of the binary iterative hard thresholding (BIHT) algorithm which is a state-of-the-art recovery algorithm in one-bit compressive sensing. The basic idea of the convergence analysis is to view BIHT as a kind of projected subgradient method under sparsity constrains. To the best of our knowledge, this is the first convergence analysis of BIHT. We first consider a general convex function subject to sparsity constraints and connect it with the non-convex model in one-bit compressive sensing literatures. A projected subgradient method is proposed to solve the general model and some convergence results are established. A stronger convergence theorem for α-strongly convex functions without assumption on differentiable condition is also established. Furthermore, the corresponding stochastic projected subgradient method is provided with convergence guarantee. In our settings, BIHT is a special case of the projected subgradient method. Therefore, the convergence analysis can be applied to BIHT naturally. Then, we apply the projected subgradient method to some related non-convex optimization models arising in compressive sensing with 11-constraint, sparse support vector machines, and rectifier linear units regression. Finally, some numerical examples are presented to show the validity of our convergence analysis. The numerical experiments also show that the proposed projected subgradient method is very simple to implement, robust to sparse noise, and effective for sparse recovery problems.

Most of the data scientists face today might be classified as multimodal data, i.e., being composed of distinct subcomponents. One common task is to separate such data into appropriate single components for further analysis. In this paper, we consider data separation from fewer, linear, nonadaptive, and noisy measurements. We show that the distinct subcomponents, which are (approximately) sparse in morphologically different (redundant) dictionaries, can be reconstructed by solving the split-analysis algorithm, provided that the dictionaries satisfy a mutual coherence (between the different dictionaries) condition and the measurement matrix satisfies a restricted isometry property adapted to a composed dictionary. These conditions impose no incoherence restriction on the dictionaries themselves, and our main result may be the first of this kind.