[论文解读] Clustering of graph vertex subset via Krylov subspace model reduction.

Vladimir Druskin, A. Mamonov+1 more|arXiv (Cornell University)|Sep 9, 2018

Model Reduction and Neural Networks被引用 1

一句话总结

本文提出两种基于Krylov子空间的模型降阶算法，用于高效地对图中目标顶点子集进行谱聚类。通过构建一个近似图拉普拉斯矩阵在目标子集上扩散转移函数的降阶模型，该方法降低了Krylov子空间的维度，使得能够使用先进的k-means近似方法实现鲁棒且可扩展的聚类，从而在显著降低计算成本的同时保持一致的聚类结果。

ABSTRACT

Clustering via graph-Laplacian spectral imbedding is ubiquitous in data science and machine learning. However, it becomes less efficient for large data sets due to two factors. First, computing the partial eigendecomposition of the graph-Laplacian typically requires a large Krylov subspace. Second, after the spectral imbedding is complete, the clustering is typically performed with various relaxations of k-means, which may become prone to getting stuck in local minima and scale poorly in terms of computational cost for large data sets. Here we propose two novel algorithms for spectral clustering of a subset of the graph vertices (target subset) based on the theory of model order reduction. They rely on realizations of a reduced order model (ROM) that accurately approximates the diffusion transfer function of the original graph for inputs and outputs restricted to the target subset. While our focus is limited to this subset, our algorithms produce its clustering that is consistent with the overall structure of the graph. Moreover, working with a small target subset reduces greatly the required dimension of Krylov subspace and allows to exploit the approximations of k-means in the regimes when they are most robust and efficient, as verified by the numerical experiments. There are several uses for our algorithms. First, they can be employed on their own to clusterize a representative subset in cases when the full graph clustering is either infeasible or not required. Second, they may be used for quality control. Third, as they drastically reduce the problem size, they enable the application of more powerful approximations of k-means like those based on semi-definite programming (SDP) instead of the conventional Lloyd's algorithm. Finally, they can be used as building blocks of a divide-and-conquer algorithm for the full graph clustering. The latter will be reported in a separate article.

研究动机与目标

• 为解决谱聚类在大规模图上因高维Krylov子空间需求而导致的计算效率低下问题。
• 通过仅聚焦于目标顶点子集而非整个图，降低聚类的计算负担。
• 通过大幅减小问题规模，使更精确且鲁棒的k-means近似方法（如基于半定规划的方法）得以应用。
• 开发一个支持质量控制的框架，可作为完整图聚类中分治算法的组成部分。

提出的方法

• 构建图拉普拉斯矩阵的降阶模型（ROM），以准确近似限制在目标顶点子集上的扩散转移函数。
• 使用Krylov子空间方法生成图拉普拉斯矩阵的低维投影，以保留与目标子集相关的谱特性。
• 利用ROM进行谱嵌入，将目标顶点映射到低维空间，进而在该空间中执行聚类。
• 利用降低的维度，应用在完整图上计算上不可行的先进k-means近似方法（例如基于SDP的方法）。
• 通过ROM公式保持目标子集与图其余部分之间的扩散动力学，确保与全局图结构的一致性。
• 利用ROM的精度，在问题规模减小的情况下仍能保持聚类质量。

实验结果

研究问题

• RQ1 能否有效将基于Krylov子空间方法的模型降阶应用于图顶点子集的谱聚类？
• RQ2 降阶模型是否保留了目标子集准确聚类所必需的谱结构？
• RQ3 所提方法是否能在保持聚类保真度的同时显著降低Krylov子空间的维度？
• RQ4 由于问题规模减小，先进k-means近似方法的应用程度如何？
• RQ5 与标准谱聚类相比，该方法在大规模图上的效率和准确性如何？

主要发现

• 所提算法通过仅聚焦于目标顶点子集，显著降低了Krylov子空间的维度，从而大幅减少了计算成本。
• 降阶模型保持了足够的精度，使聚类结果与全局图结构保持一致。
• 更小的问题规模使得能够应用更鲁棒的k-means近似方法（如半定规划），而这些方法在大规模图上原本不可行。
• 数值实验表明，该方法在效率和准确性方面表现优异，尤其在k-means松弛方法最有效的场景下。
• 该方法支持实际应用（如质量控制），并可集成到用于完整图聚类的分治策略中。