A variational approach to the consistency of spectral clustering
This provides theoretical guarantees for spectral clustering methods, addressing a foundational problem in machine learning for data analysis applications.
The paper establishes the consistency of spectral clustering for point cloud data by proving spectral convergence of graph Laplacians to continuum operators, with sharp conditions on connectivity radius scaling, and shows discrete clusters converge to a continuum partition minimizing a functional.
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence to hold. We also show that the discrete clusters obtained via spectral clustering converge towards a continuum partition of the ground truth measure. Such continuum partition minimizes a functional describing the continuum analogue of the graph-based spectral partitioning. Our approach, based on variational convergence, is general and flexible.