通过原子分解的高效且有保证的秩最小化
Efficient and Guaranteed Rank Minimization by Atomic Decomposition
AI总结：
本文提出一种基于原子分解的高效秩最小化算法，结合随机化快速低秩近似，具有性能保证且能处理大规模问题。
摘要：
Recht、Fazel 和 Parrilo 提出了秩最小化与 $\ell_0$ 范数最小化之间的类比。在秩受限等距性质下，核范数最小化是一种有保证的秩最小化算法。由此产生的半定规划是一个凸问题，但在实践中，其算法难以扩展到大规模实例。相反，我们探索了类比中缺失的项，并提出了一种新算法，该算法计算效率高且具有性能保证。该算法基于矩阵变量的原子分解，并扩展了 CoSaMP 算法在 $\ell_0$ 范数最小化中的思想。结合近期基于随机化的快速低秩矩阵近似，所提出的算法可以高效处理大规模秩最小化问题。
英文摘要：
Recht, Fazel, and Parrilo provided an analogy between rank minimization and $\ell_0$-norm minimization. Subject to the rank-restricted isometry property, nuclear norm minimization is a guaranteed algorithm for rank minimization. The resulting semidefinite formulation is a convex problem but in practice the algorithms for it do not scale well to large instances. Instead, we explore missing terms in the analogy and propose a new algorithm which is computationally efficient and also has a performance guarantee. The algorithm is based on the atomic decomposition of the matrix variable and extends the idea in the CoSaMP algorithm for $\ell_0$-norm minimization. Combined with the recent fast low rank approximation of matrices based on randomization, the proposed algorithm can efficiently handle large scale rank minimization problems.
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