Yuqiang Guan

Kernel k-means: spectral clustering and normalized cuts

Kernel k-means and spectral clustering have both been used to identify clusters that are non-linearly separable in input space. Despite significant research, these methods have remained only loosely related. In this paper, we give an explicit theoretical connection between them. We show the generality of the weighted kernel k-means objective function, and derive the spectral clustering objective of normalized cut as a special case. Given a positive definite similarity matrix, our results lead to a novel weighted kernel k-means algorithm that monotonically decreases the normalized cut. This has important implications: a) eigenvector-based algorithms, which can be computationally prohibitive, are not essential for minimizing normalized cuts, b) various techniques, such as local search and acceleration schemes, may be used to improve the quality as well as speed of kernel k-means. Finally, we present results on several interesting data sets, including diametrical clustering of large gene-expression matrices and a handwriting recognition data set.

Weighted Graph Cuts without Eigenvectors A Multilevel Approach

A variety of clustering algorithms have recently been proposed to handle data that is not linearly separable; spectral clustering and kernel k-means are two of the main methods. In this paper, we discuss an equivalence between the objective functions used in these seemingly different methods - in particular, a general weighted kernel k-means objective is mathematically equivalent to a weighted graph clustering objective. We exploit this equivalence to develop a fast high-quality multilevel algorithm that directly optimizes various weighted graph clustering objectives, such as the popular ratio cut, normalized cut, and ratio association criteria. This eliminates the need for any eigenvector computation for graph clustering problems, which can be prohibitive for very large graphs. Previous multilevel graph partitioning methods such as Metis have suffered from the restriction of equal-sized clusters; our multilevel algorithm removes this restriction by using kernel k-means to optimize weighted graph cuts. Experimental results show that our multilevel algorithm outperforms a state-of-the-art spectral clustering algorithm in terms of speed, memory usage, and quality. We demonstrate that our algorithm is applicable to large-scale clustering tasks such as image segmentation, social network analysis, and gene network analysis.

Efficient Clustering of Very Large Document Collections

An invaluable portion of scientific data occurs naturally in text form. Given a large unlabeled document collection, it is often helpful to organize this collection into clusters of related documents. By using a vector space model, text data can be treated as high-dimensional but sparse numerical data vectors. It is a contemporary challenge to efficiently preprocess and cluster very large document collections. In this paper we present a time and memory efficient technique for the entire clustering process, including the creation of the vector space model. This efficiency is obtained by (i) a memory-efficient multi-threaded preprocessing scheme, and (ii) a fast clustering algorithm that fully exploits the sparsity of the data set. We show that this entire process takes time that is linear in the size of the document collection. Detailed experimental results are presented — a highlight of our results is that we are able to effectively cluster a collection of 113,716 NSF award abstracts in 23 minutes (including disk I/O costs) on a single workstation with modest memory consumption.

A fast kernel-based multilevel algorithm for graph clustering

Graph clustering (also called graph partitioning) --- clustering the nodes of a graph --- is an important problem in diverse data mining applications. Traditional approaches involve optimization of graph clustering objectives such as normalized cut or ratio association; spectral methods are widely used for these objectives, but they require eigenvector computation which can be slow. Recently, graph clustering with a general cut objective has been shown to be mathematically equivalent to an appropriate weighted kernel k-means objective function. In this paper, we exploit this equivalence to develop a very fast multilevel algorithm for graph clustering. Multilevel approaches involve coarsening, initial partitioning and refinement phases, all of which may be specialized to different graph clustering objectives. Unlike existing multilevel clustering approaches, such as METIS, our algorithm does not constrain the cluster sizes to be nearly equal. Our approach gives a theoretical guarantee that the refinement step decreases the graph cut objective under consideration. Experiments show that we achieve better final objective function values as compared to a state-of-the-art spectral clustering algorithm: on a series of benchmark test graphs with up to thirty thousand nodes and one million edges, our algorithm achieves lower normalized cut values in 67% of our experiments and higher ratio association values in 100% of our experiments. Furthermore, on large graphs, our algorithm is significantly faster than spectral methods. Finally, our algorithm requires far less memory than spectral methods; we cluster a 1.2 million node movie network into 5000 clusters, which due to memory requirements cannot be done directly with spectral methods.

Clustering with Entropy-Like k-Means Algorithms

The aim of this chapter is to demonstrate that many results attributed to the classical k-means clustering algorithm with the squared Euclidean distance can be extended to many other distance-like functions. We focus on entropy-like distances based on Bregman [88] and Csiszar [119] divergences, which have previously been shown to be useful in various optimization and clustering contexts. Further, the chapter reviews various versions of the classical k-means and BIRCH clustering algorithms with squared Euclidean distance and considers modifications of these algorithms with the proposed families of distance-like functions. Numerical experiments with some of these modifications are reported.