SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1686-A1713, January 2020.
In this paper, we study fixed-point schemes with certain low-rank structures arising from matrix optimization problems. Traditional first-order methods depend on the eigenvalue decomposition at each iteration, which may take most of the computational time. In order to reduce the cost, we propose an inexact algorithmic framework based on a polynomial subspace extraction. The idea is to use an additional polynomial-filtered iteration to extract an approximated eigenspace and to project the iteration matrix on this subspace, followed by an optimization update. The accuracy of the extracted subspace can be controlled by the degree of the polynomial filters. This kind of subspace extraction also enjoys the warm-start property: the subspace of the current iteration is refined from the previous one. Then this framework is instantiated into two algorithms: the polynomial-filtered proximal gradient method and the polynomial-filtered alternating direction method of multipliers. The convergence of the proposed framework is guaranteed if the polynomial degree grows with an order $\mathcal{O}(\log k)$ at the $k$th iteration. If the warm-start property is considered, the degree can be reduced to a constant, independent of the iteration $k$. Preliminary numerical experiments on several matrix optimization problems show that the polynomial-filtered algorithms usually provide multifold speedups.

中文翻译：

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使用多项式滤波子空间提取的低秩矩阵迭代

SIAM科学计算杂志，第42卷，第3期，第A1686-A1713页，2020年1月。
在本文中，我们研究由矩阵优化问题引起的具有某些低秩结构的定点方案。传统的一阶方法取决于每次迭代的特征值分解，这可能会占用大部分计算时间。为了降低成本，我们提出了一种基于多项式子空间提取的不精确算法框架。想法是使用附加的多项式滤波迭代，以提取近似特征空间，并将迭代矩阵投影到此子空间上，然后进行优化更新。提取的子空间的精度可以通过多项式滤波器的程度来控制。这种子空间提取还具有热启动属性：当前迭代的子空间是从前一个子空间中提炼出来的。然后将该框架实例化为两个算法：乘数的多项式滤波近端梯度法和多项式滤波交替方向法。如果多项式度在第k次迭代时以阶数\ mathcal {O}（\ log k）$增长，则可以保证所提出框架的收敛性。如果考虑了热启动属性，则可以将度数减小为一个常数，而与迭代$ k $无关。对几个矩阵优化问题的初步数值实验表明，多项式滤波算法通常可提供倍数的加速比。如果多项式度在第k次迭代时以阶数\ mathcal {O}（\ log k）$增长，则可以保证所提出框架的收敛性。如果考虑了热启动属性，则可以将度数减小为一个常数，而与迭代$ k $无关。对几个矩阵优化问题的初步数值实验表明，多项式滤波算法通常可提供倍数的加速比。如果多项式度在第k次迭代时以阶数\ mathcal {O}（\ log k）$增长，则可以保证所提出框架的收敛性。如果考虑了热启动属性，则可以将度数减小为一个常数，而与迭代$ k $无关。对几个矩阵优化问题的初步数值实验表明，多项式滤波算法通常可提供倍数的加速比。