Spectral clustering

Given a set of data points A, the similarity matrix may be defined as a matrix ${\displaystyle S}$, where ${\displaystyle S_{ij}}$ represents a measure of the similarity between points ${\displaystyle i,j\in A}$. Spectral clustering techniques make use of the spectrum of the similarity matrix of the data to perform dimensionality reduction for clustering in fewer dimensions.

Algorithms

One such technique is the Normalized Cuts algorithm or Shi–Malik algorithm introduced by Jianbo Shi and Jitendra Malik,^[1] commonly used for image segmentation. It partitions points into two sets ${\displaystyle (S_{1},S_{2})}$  based on the eigenvector ${\displaystyle v}$  corresponding to the second-smallest eigenvalue of the Laplacian matrix

${\displaystyle L=I-D^{-1/2}SD^{-1/2}}$ 

of ${\displaystyle S}$ , where ${\displaystyle D}$  is the diagonal matrix

${\displaystyle D_{ii}=\sum _{j}S_{ij}.}$ 

This partitioning may be done in various ways, such as by taking the median ${\displaystyle m}$  of the components in ${\displaystyle v}$ , and placing all points whose component in ${\displaystyle v}$  is greater than ${\displaystyle m}$  in ${\displaystyle S_{1}}$ , and the rest in ${\displaystyle S_{2}}$ . The algorithm can be used for hierarchical clustering by repeatedly partitioning the subsets in this fashion.

A related algorithm is the Meila–Shi algorithm^[2], which takes the eigenvectors corresponding to the k largest eigenvalues of the matrix ${\displaystyle P=SD^{-1}}$  for some k, and then invokes another algorithm (e.g. k-means) to cluster points by their respective k components in these eigenvectors.

Relationship with K-Means

The Kernel K-Means problem is an extension of the K-Means problem where the input data points are mapped non-linearly into a higher-dimensional feature space via a kernel function ${\displaystyle k(x_{i},x_{j})=\phi ^{T}(x_{i})\phi (x_{j})}$ . The weighted kernel K-Means problem further extends this problem by defining a weight ${\displaystyle w_{r}}$  for each cluster,

${\displaystyle \max _{C_{i}}\sum _{r=1}^{k}w_{r}\sum _{x_{i},x_{j}\in C_{r}}k(x_{i},x_{j}).}$ 

Suppose ${\displaystyle F}$  is a matrix of the normalizing coefficients for each point for each cluster ${\displaystyle F_{ij}=w_{r}}$  if ${\displaystyle i,j\in C_{r}}$  and zero otherwise. Suppose ${\displaystyle K}$  is the kernel matrix for all points. The Weighted Kernel K-Means problem with n points and k clusters is given as,

${\displaystyle \max _{F}trace(KF)}$ 

such that,

${\displaystyle F=G_{n\times k}G_{n\times k}^{T}}$ 

${\displaystyle rank(G)=k}$ 

${\displaystyle G^{T}G=I}$ 

${\displaystyle F\cdot 1=1}$ 

${\displaystyle F^{T}1=1}$ 

This problem can be recast as,

${\displaystyle \max _{G}trace(G^{T}G).}$ 

This problem is equivalent to the Spectral Clustering problem when the identity constraints on F are relaxed.

References

1. Jianbo Shi and Jitendra Malik, "Normalized Cuts and Image Segmentation", IEEE Transactions on PAMI, Vol. 22, No. 8, Aug 2000.
2. Marina Meilă & Jianbo Shi, "Learning Segmentation by Random Walks", Processing Systems 13 (NIPS 2000), 2001, pp. 873-879.

See also

• Kernel principal component analysis
• Cluster analysis