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

arXiv: https://arxiv.org/abs/1810.10695 · HTML: https://mlsanity.com/paper/1810.10695/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2018.

## Built on by (most cited first, 6 of 6)

- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [ManiFeSt: Manifold-based Feature Selection for Small Data Sets](https://mlsanity.com/paper/2207.08574/) (2022)
- [Robust Regularized Locality Preserving Indexing for Fiedler Vector Estimation](https://mlsanity.com/paper/2107.12070/) (2021)
- [Manifold learning via quantum dynamics](https://mlsanity.com/paper/2112.11161/) (2021)
- [Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization](https://mlsanity.com/paper/2510.02308/) (2025)
- [Graph energy as a measure of community detectability in networks](https://mlsanity.com/paper/2601.05065/) (2026)

## Abstract (arXiv)

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.