Spectral Embedding Norm: Looking Deep into the Spectrum of the Graph Laplacian
This addresses a non-trivial task in image analysis, such as anomaly and target detection, with practical importance in fields like remote sensing and neuroimaging.
The paper tackles the problem of extracting clusters from datasets with multiple clusters and significant background, where traditional spectral clustering fails, by proposing the spectral embedding norm that sums squared values of normalized eigenvectors, proving it can separate clusters from background in unbalanced settings and demonstrating application on synthetic and real-world datasets.
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 $K$ eigenvectors to detect $K$ clusters, fails in such cases. In this paper we propose the {\it spectral embedding norm} which sums the squared values of the first $I$ normalized eigenvectors, where $I$ can be significantly larger than $K$. 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 $I$, and we demonstrate its application on synthetic and real-world remote sensing and neuroimaging datasets.