Spectral clustering based on local linear approximations

Ery Arias-Castro, Guangliang Chen, Gilad Lerman

Research output: Contribution to journalArticlepeer-review

54 Scopus citations

Abstract

In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a prototype for a higher-order spectral clustering method based on the residual from a local linear approximation. We obtain theoretical guarantees for this algorithm and show that, in terms of both separation and robustness to outliers, it outperforms the standard spectral clustering algorithm (based on pairwise distances) of Ng, Jordan and Weiss (NIPS'01). The optimal choice for some of the tuning parameters depends on the dimension and thickness of the clusters. We provide estimators that come close enough for our theoretical purposes. We also discuss the cases of clusters of mixed dimensions and of clusters that are generated from smoother surfaces. In our experiments, this algorithm is shown to outperform pairwise spectral clustering on both simulated and real data.

Original languageEnglish (US)
Pages (from-to)1537-1587
Number of pages51
JournalElectronic Journal of Statistics
Volume5
DOIs
StatePublished - 2011

Keywords

  • Detection of clusters in point clouds
  • Dimension estimation
  • Higher-order affinities
  • Local linear approximation
  • Local polynomial approximation
  • Nearest-neighbor search
  • Spectral clustering

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