通过半定规划进行稀疏逼近
Sparse Approximation by Semidefinite Programming
AI总结：
研究稀疏逼近问题，提出将其作为组合优化问题，基于半定规划（SDP）提供算法，有基于问题半定松弛的随机算法及去随机化算法，性能取决于字典相关常数。
摘要：
在过去十年中，稀疏逼近问题以及与之密切相关的压缩感知受到了极大关注。主要从应用谐波分析和信号处理的角度进行研究，针对该问题有两种主要算法方法：称为匹配追踪（MP）的贪心方法和称为基追踪（BP）的基于线性规划的方法。本文旨在通过将稀疏逼近视为组合优化问题并提供基于强大优化技术半定规划（SDP）的算法，为其带来新视角。特别地，我们展示了一种基于问题半定松弛的随机算法，其性能保证取决于所用字典的相干性和受限等距常数。然后基于条件概率方法展示了该算法的去随机化。
英文摘要：
The problem of sparse approximation and the closely related compressed sensing have received tremendous attention in the past decade. Primarily studied from the viewpoint of applied harmonic analysis and signal processing, there have been two dominant algorithmic approaches to this problem: Greedy methods called the matching pursuit (MP) and the linear programming based approaches called the basis pursuit (BP). The aim of the current paper is to bring a fresh perspective to sparse approximation by treating it as a combinatorial optimization problem and providing an algorithm based on the powerful optimization technique semidefinite programming (SDP). In particular, we show that there is a randomized algorithm based on a semidefinite relaxation of the problem with performance guarantees depending on the coherence and the restricted isometry constant of the dictionary used. We then show a derandomization of the algorithm based on the method of conditional probabilities.
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