A low-rank ODE for spectral clustering stability

Spectral clustering is a well-known technique which identifies k clusters in an undirected graph, with n vertices and weight matrix , by exploiting its graph Laplacian . In particular, the k clusters can be identified by the knowledge of the eigenvectors associated with the smallest non zero eigenvalues of , say (recall that ). Identifying k is an essential task of a clustering algorithm, since if is close to the reliability of the method is reduced. The k-th spectral gap is often considered as a stability indicator. This difference can be seen as an unstructured distance between and an arbitrary symmetric matrix with vanishing k-th spectral gap. A more appropriate structured distance to ambiguity such that represents the Laplacian of a graph has been proposed in Andreotti et al. (2021) [2]. This is defined as the minimal distance between 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.