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.

中文翻译：

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基联合中稀疏表示的一个开放问题

我们考虑来自冗余字典的信号的稀疏表示，这些字典是几个正交基的联合。Donoho 和 Elad 引入的 spark 在稀疏表示中起着重要作用。然而，火花的数值计算通常是组合的。对于几个正交基的并集，之前的工作通过相互相干建立了火花的两个下界。我们建设性地证明它们都是紧的。我们的主要结果对 Gribonval 和 Nielsen 在正交基的联合中的稀疏表示的开放问题给出了肯定的答案。构造性证明依赖于一系列相互无偏的基础，这些基础首次出现在量子信息理论中。