Practical Algorithms of Spectral Clustering: Toward Large-Scale Vision-Based Motion Analysis

Abstract

This chapter presents some practical algorithms of spectral clustering for large-scale data. Spectral clustering is a kernel-based method of grouping data on separate nonlinear manifolds. Reducing its computational expense without critical loss of accuracy contributes to its practical use especially in vision-based applications. The present algorithms exploit random projection and subsampling techniques for reducing dimensionality and the cost for evaluating pairwise similarities of data. The computation time is quasilinear with respect to the data cardinality, and it can be independent of data dimensionality in some appearance-based applications. The efficiency of the algorithms is demonstrated in appearance-based image/video segmentation.

Appendix: Clustering Scores

The conditional entropy (CE) and the normalized mutual information (NMI) [27] are defined as follows.

$$ \begin{array}{*{20}c} {{\rm CE} = \mathop \Sigma \limits_{i = 1}^k \frac{{\left| {C_i } \right|}}{{ - n\log k}}\mathop \Sigma \limits_{j = 1}^k \frac{{\left| {X_{ij} } \right|}}{{\left| {C_i } \right|}}\log \frac{{\left| X \right|}}{{\left| {C_i } \right|}},}\\ {{\rm NMI} = \frac{{\Sigma _{i = 1}^k \Sigma _{j = 1}^k \frac{{\left| {X_{ij} } \right|}}{n}\log \frac{{n\left| {X_{ij} } \right|}}{{\left| {C_i \left| {A_j } \right|} \right|}}}}{{\sqrt {(\Sigma _{i = 1}^k \frac{{\left| {C_i } \right|}}{n}\log } \frac{{\left| {C_i } \right|}}{n})(\Sigma _{j = 1}^k \frac{{\left| {A_j } \right|}}{n}\log \frac{{\left| {A_j } \right|}}{n})}}}\\ \end{array}$$

Here, |C _i | and |A _j | are the numbers of samples in the estimated cluster C _i and the optimal cluster A _j , respectively. X _ij =C _i ∩A _j is the set of common samples. The smaller the CE is, or the larger the NMI is, the better the clustering result is. The NMI takes a value between 0 and 1.