发表机构
Department of Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium; Department of Mathematics, University of British Columbia, Vancouver, BC, Canada（荷语鲁汶大学电气工程系（ESAT）; 不列颠哥伦比亚大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出将欠定线性系统的稀疏逼近问题转化为多项式方程组，并开发了基于特征值分解及两种优化方法，可显式控制稀疏度并保证恢复所有稀疏解，实验验证了其优于基追踪和正交匹配追踪的性能。
AI 中文摘要
我们考虑寻找欠定线性系统$Ax=b$的稀疏解的问题。与基于贪婪算法或凸松弛的传统方法不同，我们将稀疏逼近重新表述为一个结构化的多项式方程组，并与张量方法的相关文献建立联系。我们开发了一种基于特征值分解的方法，该方法在存在多个稀疏解时能形式化地保证恢复所有稀疏解。我们还开发了两种基于优化的方法，实现了有利的计算复杂度。新方法允许对目标稀疏度进行显式控制。数值实验展示了其性能，并与基追踪（去噪）和正交匹配追踪进行了比较。
英文摘要
We consider the problem of finding sparse solutions of an underdetermined linear system $Ax=b$. In contrast to conventional approaches based on greedy algorithms or convex relaxation, we reformulate sparse approximation as a structured system of polynomial equations and connect with the literature on tensor methods. We develop an eigenvalue decomposition based method that formally guarantees recovery of all sparse solutions if there is more than one. We also develop two optimization-based methods achieving favorable computational complexity. The new methods allow explicit control of the target sparsity. Numerical experiments illustrate the performance and compare to basis pursuit (denoising) and orthogonal matching pursuit.
Comments10 pages, 9 figures, 1 table. Submitted to IEEE Transactions on Signal Processing
