Convex programming based spectral clustering

Abstract

Clustering is a fundamental task in data analysis, and spectral clustering has been recognized as a promising approach to it. Given a graph describing the relationship between data, spectral clustering explores the underlying cluster structure in two stages. The first stage embeds the nodes of the graph in real space, and the second stage groups the embedded nodes into several clusters. The use of the k-means method in the grouping stage is currently standard practice. We present a spectral clustering algorithm that uses convex programming in the grouping stage and study how well it works. This algorithm is designed based on the following observation. If a graph is well-clustered, then the nodes with the largest degree in each cluster can be found by computing an enclosing ellipsoid of the nodes embedded in real space, and the clusters can be identified by using those nodes. We show that, for well-clustered graphs, the algorithm can find clusters of nodes with minimal conductance. We also give an experimental assessment of the algorithm's performance.

Multiway p-spectral graph cuts on Grassmann manifolds

“…As opposed to other popular clustering techniques, such as the k-means (MacQueen, 1967 ) and the expectation-maximization algorithm (Dempster et al., 1977 ), spectral methods perform well in nonconvex sample spaces, as they can avoid local minima (Bichot & Siarry, 2013 ). They have therefore been successfully applied in various fields of data clustering, such as computer vision (Malik et al., 2001 ), load balancing (Hendrickson & Leland, 1995 ), biological systems (Pentney & Meila, 2005 ) and text classification (Aggarwal & Zhai, 2012 ), and are a field of active research (Ge et al., 2021 ; Mizutani, 2021 ). Additionally, efficient variants employing multilevel techniques have been proposed (Dhillon et al., 2005 , 2007 ).…”