Frobenius范数下的最近图拉普拉斯矩阵
The Nearest Graph Laplacian in Frobenius Norm
AI总结：
针对Frobenius范数下给定矩阵的最近有向图拉普拉斯矩阵问题，提出基于积极集和KKT点直接计算的两种凸二次优化算法，具有更低复杂度与有限终止性。
摘要：
我们研究寻找给定矩阵在Frobenius范数距离度量下最近图拉普拉斯矩阵的问题。具体而言，针对有向图拉普拉斯矩阵，我们通过将该问题重构为具有特殊结构的凸二次优化问题，提出了两种新算法：一种基于积极集方法，另一种基于Karush-Kuhn-Tucker（KKT）点的直接计算。所提出的算法可应用于涉及拉普拉斯动力学的系统辨识和模型降阶问题。我们证明，与内点法和V-FISTA（后者是一种加速投影梯度法）不同，这些算法具有更低的时间复杂度和有限终止性质。数值实验证实了所提算法的有效性。
英文摘要：
We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush-Kuhn-Tucker (KKT) points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V-FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms.
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