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Spectral Embedding Norm: Looking Deep into the Spectrum of the Graph Laplacian

Abstract

The extraction of clusters from a dataset which includes multiple clusters and a significant background component is a non-trivial task of practical importance. In image analysis this manifests for example in anomaly detection and target detection. The traditional spectral clustering algorithm, which relies on the leading KK eigenvectors to detect KK clusters, fails in such cases. In this paper we propose the {\it spectral embedding norm} which sums the squared values of the first II normalized eigenvectors, where II can be significantly larger than KK. We prove that this quantity can be used to separate clusters from the background in unbalanced settings, including extreme cases such as outlier detection. The performance of the algorithm is not sensitive to the choice of II, and we demonstrate its application on synthetic and real-world remote sensing and neuroimaging datasets.

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