Connected Subspace Clustering Solves Spatially Coherent Data Grouping

Researchers have defined and addressed a new computational challenge called Connected Subspace Clustering. This problem involves partitioning high-dimensional data points, which are associated with a connectivity graph, into a specified number of connected clusters. The objective is to minimize the total squared distance of points to their respective cluster's best-fit affine subspace, ensuring both internal similarity and physical coherence within clusters. The problem is proven to be NP-hard, even in simplified scenarios. To tackle this complexity, the study proposes an efficient Lloyd-style heuristic. This method iteratively alternates between fitting subspaces to clusters and merging procedures to enforce connectivity, guaranteeing the formation of exactly k connected regions. The approach significantly outperforms unconstrained clustering methods, which often result in numerous disconnected fragments. Applied to global sea level time series, the heuristic successfully isolates signals aligned with climate indices, demonstrating its practical utility beyond its initial motivation in geodesy.