关于允许泛稀疏表示的最小过完备性
On the Minimal Overcompleteness Allowing Universal Sparse Representation
摘要：
冗余字典上的稀疏表示构成了许多类信号的良好模型（例如，自然图像块、语音信号段等）。然而，尽管它的流行，很少有人知道这个模型的代表性能力。在本文中，我们研究如何冗余的字典必须允许任何向量承认稀疏近似与规定的稀疏性和规定的精度水平。我们解决这个问题，在最坏的情况下，并在一个平均情况下。对于每一种情况下，我们推导出最小所需的过完备性的下限和上限。我们的边界有简单的封闭形式的表达式，可以很容易地推导出大尺寸的渐近行为。特别是，我们发现，所需的过完备性指数增长的稀疏性水平和多项式与允许的表示误差。这意味着通用稀疏表示仅在中等稀疏水平下是实用的，但可以在相对高的精度下实现。作为我们分析的一个副作用，我们得到了正则化不完全beta函数的一个紧下界，这本身可能很有趣。我们说明了我们的结果的有效性，通过数值模拟，支持我们的研究结果。
英文摘要：
Sparse representation over redundant dictionaries constitutes a good model for many classes of signals (e.g., patches of natural images, segments of speech signals, etc.). However, despite its popularity, very little is known about the representation capacity of this model. In this paper, we study how redundant a dictionary must be so as to allow any vector to admit a sparse approximation with a prescribed sparsity and a prescribed level of accuracy. We address this problem both in a worst-case setting and in an average-case one. For each scenario we derive lower and upper bounds on the minimal required overcompleteness. Our bounds have simple closed-form expressions that allow to easily deduce the asymptotic behavior in large dimensions. In particular, we find that the required overcompleteness grows exponentially with the sparsity level and polynomially with the allowed representation error. This implies that universal sparse representation is practical only at moderate sparsity levels, but can be achieved at relatively high accuracy. As a side effect of our analysis, we obtain a tight lower bound on the regularized incomplete beta function, which may be interesting in its own right. We illustrate the validity of our results through numerical simulations, which support our findings.
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