Inderjit S. Dhillon

Information-theoretic metric learning

In this paper, we present an information-theoretic approach to learning a Mahalanobis distance function. We formulate the problem as that of minimizing the differential relative entropy between two multivariate Gaussians under constraints on the distance function. We express this problem as a particular Bregman optimization problem---that of minimizing the LogDet divergence subject to linear constraints. Our resulting algorithm has several advantages over existing methods. First, our method can handle a wide variety of constraints and can optionally incorporate a prior on the distance function. Second, it is fast and scalable. Unlike most existing methods, no eigenvalue computations or semi-definite programming are required. We also present an online version and derive regret bounds for the resulting algorithm. Finally, we evaluate our method on a recent error reporting system for software called Clarify, in the context of metric learning for nearest neighbor classification, as well as on standard data sets.

Co-clustering documents and words using bipartite spectral graph partitioning

Both document clustering and word clustering are well studied problems. Most existing algorithms cluster documents and words separately but not simultaneously. In this paper we present the novel idea of modeling the document collection as a bipartite graph between documents and words, using which the simultaneous clustering problem can be posed as a bipartite graph partitioning problem. To solve the partitioning problem, we use a new spectral co-clustering algorithm that uses the second left and right singular vectors of an appropriately scaled word-document matrix to yield good bipartitionings. The spectral algorithm enjoys some optimality properties; it can be shown that the singular vectors solve a real relaxation to the NP-complete graph bipartitioning problem. We present experimental results to verify that the resulting co-clustering algorithm works well in practice.

Concept Decompositions for Large Sparse Text Data Using Clustering

Unlabeled document collections are becoming increasingly common and available; mining such data sets represents a major contemporary challenge. Using words as features, text documents are often represented as high-dimensional and sparse vectors–a few thousand dimensions and a sparsity of 95 to 99% is typical. In this paper, we study a certain spherical k-means algorithm for clustering such document vectors. The algorithm outputs k disjoint clusters each with a concept vector that is the centroid of the cluster normalized to have unit Euclidean norm. As our first contribution, we empirically demonstrate that, owing to the high-dimensionality and sparsity of the text data, the clusters produced by the algorithm have a certain “fractal-like” and “self-similar” behavior. As our second contribution, we introduce concept decompositions to approximate the matrix of document vectors; these decompositions are obtained by taking the least-squares approximation onto the linear subspace spanned by all the concept vectors. We empirically establish that the approximation errors of the concept decompositions are close to the best possible, namely, to truncated singular value decompositions. As our third contribution, we show that the concept vectors are localized in the word space, are sparse, and tend towards orthonormality. In contrast, the singular vectors are global in the word space and are dense. Nonetheless, we observe the surprising fact that the linear subspaces spanned by the concept vectors and the leading singular vectors are quite close in the sense of small principal angles between them. In conclusion, the concept vectors produced by the spherical k-means algorithm constitute a powerful sparse and localized “basis” for text data sets.

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Information-theoretic co-clustering

Two-dimensional contingency or co-occurrence tables arise frequently in important applications such as text, web-log and market-basket data analysis. A basic problem in contingency table analysis is co-clustering: simultaneous clustering of the rows and columns. A novel theoretical formulation views the contingency table as an empirical joint probability distribution of two discrete random variables and poses the co-clustering problem as an optimization problem in information theory---the optimal co-clustering maximizes the mutual information between the clustered random variables subject to constraints on the number of row and column clusters. We present an innovative co-clustering algorithm that monotonically increases the preserved mutual information by intertwining both the row and column clusterings at all stages. Using the practical example of simultaneous word-document clustering, we demonstrate that our algorithm works well in practice, especially in the presence of sparsity and high-dimensionality.

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.

A Data-Clustering Algorithm on Distributed Memory Multiprocessors

To cluster increasingly massive data sets that are common today in data and text mining, we propose a parallel implementation of the k-means clustering algorithm based on the message passing model. The proposed algorithm exploits the inherent data-parallelism in the kmeans algorithm. We analytically show that the speedup and the scaleup of our algorithm approach the optimal as the number of data points increases. We implemented our algorithm on an IBM POWERparallel SP2 with a maximum of 16 nodes. On typical test data sets, we observe nearly linear relative speedups, for example, 15.62 on 16 nodes, and essentially linear scaleup in the size of the data set and in the number of clusters desired. For a 2 gigabyte test data set, our implementation drives the 16 node SP2 at more than 1.8 gigaflops.

Designing structured tight frames via an alternating projection method

Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm.

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.