Despite many empirical successes of spectral clustering methods— algorithms that cluster points using eigenvectors of matrices derived from the data—there are several unresolved issues. First. there are a wide variety of algorithms that use the eigenvectors in slightly different ways. Second, many of these algorithms have no proof that they will actually compute a reasonable clustering. In this paper, we present a simple spectral clustering algorithm that can be implemented using a few lines of Matlab. Using tools from matrix perturbation theory, we analyze the algorithm, and give conditions under which it can be expected to do well. We also show surprisingly good experimental results on a number of challenging clustering problems.

/pdf/on-spectral-clustering-analysis-and-an-algorithm-w9vnbkmp9q.pdf

On Spectral Clustering: Analysis and an algorithm

Data analysis plays an indispensable role for understanding various phenomena. Cluster analysis, primitive exploration with little or no prior knowledge, consists of research developed across a wide variety of communities. The diversity, on one hand, equips us with many tools. On the other hand, the profusion of options causes confusion. We survey clustering algorithms for data sets appearing in statistics, computer science, and machine learning, and illustrate their applications in some benchmark data sets, the traveling salesman problem, and bioinformatics, a new field attracting intensive efforts. Several tightly related topics, proximity measure, and cluster validation, are also discussed.

/pdf/survey-of-clustering-algorithms-6zezqhbjyo.pdf

Survey of clustering algorithms

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

/pdf/finding-community-structure-in-networks-using-the-4em5so8glh.pdf

Finding community structure in networks using the eigenvectors of matrices

This paper introduces the problem of combining multiple partitionings of a set of objects into a single consolidated clustering without accessing the features or algorithms that determined these partitionings. We first identify several application scenarios for the resultant 'knowledge reuse' framework that we call cluster ensembles. The cluster ensemble problem is then formalized as a combinatorial optimization problem in terms of shared mutual information. In addition to a direct maximization approach, we propose three effective and efficient techniques for obtaining high-quality combiners (consensus functions). The first combiner induces a similarity measure from the partitionings and then reclusters the objects. The second combiner is based on hypergraph partitioning. The third one collapses groups of clusters into meta-clusters which then compete for each object to determine the combined clustering. Due to the low computational costs of our techniques, it is quite feasible to use a supra-consensus function that evaluates all three approaches against the objective function and picks the best solution for a given situation. We evaluate the effectiveness of cluster ensembles in three qualitatively different application scenarios: (i) where the original clusters were formed based on non-identical sets of features, (ii) where the original clustering algorithms worked on non-identical sets of objects, and (iii) where a common data-set is used and the main purpose of combining multiple clusterings is to improve the quality and robustness of the solution. Promising results are obtained in all three situations for synthetic as well as real data-sets.

/pdf/cluster-ensembles-a-knowledge-reuse-framework-for-combining-3qjcysq4q0.pdf

Cluster ensembles --- a knowledge reuse framework for combining multiple partitions

This is a survey on graph visualization and navigation techniques, as used in information visualization. Graphs appear in numerous applications such as Web browsing, state-transition diagrams, and data structures. The ability to visualize and to navigate in these potentially large, abstract graphs is often a crucial part of an application. Information visualization has specific requirements, which means that this survey approaches the results of traditional graph drawing from a different perspective.

/pdf/graph-visualization-and-navigation-in-information-24tisrjfna.pdf

Graph visualization and navigation in information visualization: A survey

Recent directions in netlist partitioning: a survey

From 1985-1993, the MCNC regularly introduced and maintained circuit benchmarks for use by the Design Automation community. However, during the last five years, no new circuits have been introduced that can be used for developing fundamental physical design applications, such as partitioning and placement. The largest circuit in the existing set of benchmark suites has over 100,000 modules, but the second largest has just over 25,000 modules, which is small by today's standards. This paper introduces the ISPD98 benchmark suite which consists of 18 circuits with sizes ranging from 13,000 to 210,000 modules. Experimental results for three existing partitioners are presented so that future researchers in partitioning can more easily evaluate their heuristics.

/pdf/the-ispd98-circuit-benchmark-suite-3o8qxmq5ry.pdf

The ISPD98 circuit benchmark suite

A global clock distribution strategy used on several microprocessor chips is described. The clock network consists of buffered tunable trees or treelike networks, with the final level of trees all driving a single common grid covering most of the chip. This topology combines advantages of both trees and grids. A new tuning method was required to efficiently tune such a large strongly connected interconnect network consisting of up to 6 m of wire and modeled with 50000 resistors, capacitors, and inductors. Variations are described to handle different floor-planning styles. Global clock skew as low as 22 ps on large microprocessor chips has been measured.

A clock distribution network for microprocessors

A spectral partitioning method uses the eigenvectors of a graph's adjacency or Laplacian matrix to construct a geometric representation (e.g., a linear ordering) which is then heuristically partitioned. We map each graph vertex to a vector in d-dimensional space, where d is the number of eigenvectors, such that these vectors constitute an instance of the vector partitioning problem. When all the eigenvectors are used, graph partitioning exactly reduces to vector partitioning. This result motivates a simple ordering heuristic that can be used to yield high-quality 2-way and multi-way partitionings. Our experiments suggest the vector partitioning perspective opens the door to new and effective heuristics.

Spectral Partitioning: The More Eigenvectors, The Better

Buffer insertion seeks to place buffers on the wires of a signal netto minimize delay. Van Ginneken [Buffer Placement in Distributed RC-tree Networks for Minimal Elmore Delay] proposed an optimal dynamicprogramming solution (with extensions proposed by [7] [8][9] [12]) such that at most one buffer can be placed on a singlewire. This constraint can hurt solution quality, but it may be circumventedby dividing each wire into multiple smaller segments.This work studies the problem of finding the correct number of segmentsfor each wire in the routing tree. Too few segments yieldssub-par solutions, but too many segments can lead to excessive runtimes and memory loads. We derive new theoretical results forcomputing the appropriate number of buffers (and hence wire segments)which motivate our new wire segmenting algorithm. Weshow that using wire segmenting as a precursor to buffer insertionproduces solutions within a few percent of optimal, while usingonly seconds of CPU time.

http://www.cecs.uci.edu/~papers/compendium94-03/papers/1997/dac97/pdffiles/37_1.pdf

Charles J. Alpert

Papers

Recent directions in netlist partitioning: a survey

The ISPD98 circuit benchmark suite

A clock distribution network for microprocessors

Spectral Partitioning: The More Eigenvectors, The Better

Wire segmenting for improved buffer insertion