Convex Programming Based Spectral Clustering

TL;DR

This paper introduces a spectral clustering algorithm that employs convex programming in the grouping stage, effectively identifying well-clustered graph structures by finding nodes with maximal degrees through ellipsoid computations.

Contribution

It proposes a novel convex programming-based spectral clustering method that improves cluster detection for well-structured graphs, moving beyond traditional k-means approaches.

Findings

The algorithm can identify clusters with minimal conductance in well-clustered graphs.

Experimental results demonstrate effective performance of the proposed method.

The approach offers a new perspective on spectral clustering using convex optimization.

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.