稀疏逼近在相干字典下可证困难
Sparse Approximation is Provably Hard under Coherent Dictionaries
摘要：
众所周知，稀疏逼近问题在一般字典下是\textsf{NP} -困难的。在过去的十年中，在对由$M \times N$矩阵表示的字典的\emph{coherence} $μ$的各种假设下，已经设计和分析了几种算法，从该矩阵中选择$k$列向量的子集。所有这些结果都假定为$μ=O(k^{-1})$。本文试图弥补一般字典下\textsf{NP} -硬度的否定结果与这种限制性假设下的肯定结果之间的巨大差距。特别是，它表明，上述假设可能是渐近的最好的一个可以达到任何有效的算法结果下，著名的复杂性理论。在建立的结果，我们利用一个新的简单的多层PCP，这是量身定制的矩阵与小的一致性结合我们的减少。
英文摘要：
It is well known that sparse approximation problem is \textsf{NP}-hard under general dictionaries. Several algorithms have been devised and analyzed in the past decade under various assumptions on the \emph{coherence} $μ$ of the dictionary represented by an $M \times N$ matrix from which a subset of $k$ column vectors is selected. All these results assume $μ=O(k^{-1})$. This article is an attempt to bridge the big gap between the negative result of \textsf{NP}-hardness under general dictionaries and the positive results under this restrictive assumption. In particular, it suggests that the aforementioned assumption might be asymptotically the best one can make to arrive at any efficient algorithmic result under well-known conjectures of complexity theory. In establishing the results, we make use of a new simple multilayered PCP which is tailored to give a matrix with small coherence combined with our reduction.
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