A low-rank ODE for spectral clustering stability

Abstract

Spectral clustering is a well-known technique which identifies k clusters in an undirected graph, with n vertices and weight matrix W∈Rn×n, by exploiting its graph Laplacian L(W). In particular, the k clusters can be identified by the knowledge of the eigenvectors associated with the k−1 smallest non zero eigenvalues of L(W), say λ2,…,λk (recall that λ1=0). Identifying k is an essential task of a clustering algorithm, since if λk+1 is close to λk the reliability of the method is reduced. The k-th spectral gap λk+1−λk is often considered as a stability indicator. This difference can be seen as an unstructured distance between L(W) and an arbitrary symmetric matrix L⋆ with vanishing k-th spectral gap. A more appropriate structured distance to ambiguity such that L⋆ represents the Laplacian of a graph has been proposed in Andreotti et al. (2021) [2]. This is defined as the minimal distance between L(W) and Laplacians of graphs with the same vertices and edges, but with weights that are perturbed such that the k-th spectral gap vanishes. In this article we consider a slightly different approach, still based on the reformulation of the problem into the minimization of a suitable functional in the eigenvalues. After determining the gradient system associated with this functional, we introduce a low-rank projected system, suggested by the underlying low-rank structure of the extremizers of the problem. The integration of this low-rank system, requires both a moderate computational effort and a memory requirement, as it is shown in some illustrative numerical examples.