稀疏近似问题：快速模拟退火的成功与失败
Sparse approximation problem: how rapid simulated annealing succeeds and fails
摘要：
基于稀疏性的信息处理技术已经在多个学科中得到了积极的研究。其中，通过过完备基的少量基向量的组合来近似表示给定数据集的数学框架被称为{ \em稀疏近似}。在本文中，我们应用模拟退火，一般优化问题的元启发式算法，稀疏近似的情况下，给定的数据有一个种植稀疏表示和噪声存在。在无噪声情况下的结果表明，我们的模拟退火工程以及在一个合理的参数区域：种植的解决方案是相当迅速地找到。即使在稀疏近似问题的常见松弛（$\ell_1$ -松弛）无效的情况下也是如此。另一方面，当数据的维数接近非零分量的数量时，另一个亚稳态出现，我们的算法无法找到种植解决方案。这种现象与一阶相变有关。在非常强的噪声的情况下，寻找种植的解决方案不再有意义。在这种情况下，我们的算法可以相当快地确定接近最小失真的解决方案。
英文摘要：
Information processing techniques based on sparseness have been actively studied in several disciplines. Among them, a mathematical framework to approximately express a given dataset by a combination of a small number of basis vectors of an overcomplete basis is termed the {\em sparse approximation}. In this paper, we apply simulated annealing, a metaheuristic algorithm for general optimization problems, to sparse approximation in the situation where the given data have a planted sparse representation and noise is present. The result in the noiseless case shows that our simulated annealing works well in a reasonable parameter region: the planted solution is found fairly rapidly. This is true even in the case where a common relaxation of the sparse approximation problem, the $\ell_1$-relaxation, is ineffective. On the other hand, when the dimensionality of the data is close to the number of non-zero components, another metastable state emerges, and our algorithm fails to find the planted solution. This phenomenon is associated with a first-order phase transition. In the case of very strong noise, it is no longer meaningful to search for the planted solution. In this situation, our algorithm determines a solution with close-to-minimum distortion fairly quickly.
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