连通子空间聚类：难度、可扩展启发式算法及在海平面大地测量中的应用

中文摘要 AI 辅助

约束优化通过整合辅助信息扩展了经典优化，使其广泛应用于科学与工程领域。考虑在不同物理位置测量变量的场景，对这些测量值分组时，通常需要簇既内部相似又符合物理一致性，因此存在一个约束聚类问题，其中约束用于建模物理一致性。受大地测量应用的启发，该应用需识别海面连续区域以进行主成分分析，我们提出连通子空间聚类问题：给定高维点和一个连通图，将其划分为k个连通簇，最小化它们到簇的最佳拟合m'维仿射子空间的总平方距离。我们证明，即使对于m'=0且带有空洞的网格图，该问题也难以在Ω(n^{1/2-ε})的近似比内求解，其中ε>0，n为测量值数量。随后，我们提出一种高效的Lloyd型启发式算法，该算法交替进行子空间拟合与迭代合并过程以强制连通性。我们的方法通过构造恰好返回k个连通区域，而非约束方法会留下多达1966个不连通片段且成本更高。在对全球海平面时间序列的160种配置研究中，我们基于合并的修复策略在73.75%的案例中是四种策略中表现最强的，且在所有测试的簇数量下始终优于（连通）Ward's方法等竞争对手。所得区域分离出与气候指数（如厄尔尼诺-南方涛动和印度洋偶极子）一致的信号。尽管为大地测量开发，该方法也适用于其他空间嵌入的多变量时间序列，如气候场、遥感、神经成像和传感器网络。

英文摘要

Constrained optimization extends classical optimization by integrating side information, making it widely applicable across scientific and engineering domains. Consider a setting where we measure variables at different physical locations. When grouping these measurements, we often want clusters that are both internally similar and physically coherent. Thus, we have a constrained clustering problem where the constraint models coherence. Motivated by an application in geodesy, where contiguous regions of the sea surface must be identified for principal component analysis, we introduce the Connected Subspace Clustering problem: given high-dimensional points and a connectivity graph, partition them into $k$ connected clusters, minimizing their total squared distance to the clusters' best-fit $m'$-dimensional affine subspaces. We prove that, even for $m' = 0$ and a grid graph with holes, the problem is NP-hard to approximate within $Ω(n^{1/2-\varepsilon})$ for every $\varepsilon>0$, where $n$ is the number of measurements. We then introduce an efficient Lloyd-style heuristic that alternates subspace fitting with an iterative merging procedure to enforce connectivity. Our method returns exactly $k$ connected regions by construction, whereas unconstrained methods leave up to $1{,}966$ disconnected fragments at higher cost. In a study of 160 configurations on global sea level time series, our merging-based repair is the strongest of four strategies in $73.75\%$ of cases, and consistently outperforms competitors such as (connected) Ward's method across all tested cluster counts. The resulting regions isolate signals aligning with climate indices such as the El Nino-Southern Oscillation and Indian Ocean Dipole. Although developed for geodesy, the approach applies to other spatially embedded multivariate time series, such as climate fields, remote sensing, neuroimaging, and sensor networks.

发表机构

• Heinrich Heine University Düsseldorf（海因里希·海涅杜塞尔多夫大学）
• University of Cologne（科隆大学）
• University of Bonn（波恩大学）
• University at Buffalo（纽约州立大学布法罗分校）

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