Hierarchical Clustering via Spectral Approach

Spectral clustering is an unsupervised learning method, renowned for its ability to identify non-convex structures by leveraging the eigenstructure of graph Laplacians. Despite its efficacy, the classical framework is designed to yield a single partition and requires the pre-specification of the number of clusters and the embedding dimension. In cluster analysis, however, the interest often lies in recovering a hierarchy of nested groups rather than a single partition, motivating the use of hierarchical spectral approaches. In this paper, we propose hierarchical spectral clustering, a recur-sive top-down framework designed to generate a representation of the data without a priori parametric assumptions. Unlike existing hierarchi-cal variants, hierarchical spectral clustering adopts a dynamic splitting criterion: at each iteration, the algorithm identifies the cluster with the minimum algebraic connectivity–measured by the second smallest eigen-value of the local normalized Laplacian–and performs a bipartite split using the corresponding Fiedler vector. By recomputing the Laplacian operator locally at every level, the method adaptively captures the intrin-sic topology of each sub-component. This approach produces a dendrogram where the hierarchy is deter-mined by a graph-theoretic criterion capturing how well a subset of nodes is separated from the rest of the graph, in accordance with the Cheeger inequality. Experiments on synthetic datasets show that the proposed hierarchical spectral clustering successfully reveals cluster structures at different levels, providing a hierarchy of nested solutions rather than a single partition.