Spectral clustering based on local linear approximations

TL;DR

This paper introduces a higher-order spectral clustering method based on local linear approximations, which improves robustness and separation in clustering tasks with outliers and complex structures.

Contribution

It proposes a novel spectral clustering approach using residuals from local linear approximations, with theoretical guarantees and superior performance over standard methods.

Findings

Outperforms standard spectral clustering in robustness and separation

Provides estimators for tuning parameters based on cluster properties

Demonstrates effectiveness on simulated and real datasets

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.