Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling

TL;DR

This paper introduces the TSCC algorithm for Hybrid Linear Modeling, providing theoretical analysis that guarantees effective clustering of data from mixtures of affine subspaces under certain conditions.

Contribution

It proposes the TSCC algorithm and offers rigorous theoretical justification for its performance in segmenting data into affine subspace clusters.

Findings

TSCC effectively segments data from mixtures of affine subspaces.

Clustering quality depends on within-cluster errors and between-cluster interactions.

The analysis offers new insights into spectral clustering methods.

Abstract

The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algorithm for solving the HLM problem, and provides careful analysis to justify it. The TSCC algorithm is practically a combination of Govindu's multi-way spectral clustering framework (CVPR 2005) and Ng et al.'s spectral clustering algorithm (NIPS 2001). The main result of this paper states that if the given data is sampled from a mixture of distributions concentrated around affine subspaces, then with high sampling probability the TSCC algorithm segments well the different underlying clusters. The goodness of clustering depends on the within-cluster errors, the between-clusters interaction, and a tuning parameter applied by TSCC. The proof also provides new insights for the analysis of Ng et al. (NIPS 2001).