A Parallel Projection Method for Metric Constrained Optimization

Ruggles, Cameron; Veldt, Nate; Gleich, David F.

doi:10.1137/1.9781611976229.5

Cited by 4 publications

(4 citation statements)

References 27 publications

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“…Computing the LambdaCC linear programming relaxation can be challenging due to the size of the constraint set. For our smaller graphs we apply Gurobi optimization software, and for larger problems we use recently developed memory-efficient projection methods [37,31]. For the local-flow objective we use a fast Julia implementation we developed in recent work [39].…”

Section: Methodsmentioning

confidence: 99%

“…We note for example that the resolution parameter corresponding to modularity is λ = 1/(2|E|), which is also inversely proportional to |E|. Computing all of the LP bounds is the bottleneck in our computations, and takes just under 2.5 hours using a recently developed parallel solver for the correlation clustering relaxation [31].…”

Section: Meta-data and Global Clusteringmentioning

confidence: 99%

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Learning Resolution Parameters for Graph Clustering

Veldt

Gleich

Wirth

2019

The World Wide Web Conference

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Finding clusters of well-connected nodes in a graph is an extensively studied problem in graph-based data analysis. Because of its many applications, a large number of distinct graph clustering objective functions and algorithms have already been proposed and analyzed. To aid practitioners in determining the best clustering approach to use in different applications, we present new techniques for automatically learning how to set clustering resolution parameters. These parameters control the size and structure of communities that are formed by optimizing a generalized objective function. We begin by formalizing the notion of a parameter fitness function, which measures how well a fixed input clustering approximately solves a generalized clustering objective for a specific resolution parameter value. Under reasonable assumptions, which suit two key graph clustering applications, such a parameter fitness function can be efficiently minimized using a bisection-like method, yielding a resolution parameter that fits well with the example clustering. We view our framework as a type of single-shot hyperparameter tuning, as we are able to learn a good resolution parameter with just a single example. Our general approach can be applied to learn resolution parameters for both local and global graph clustering objectives. We demonstrate its utility in several experiments on real-world data where it is helpful to learn resolution parameters from a given example clustering.

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Section: Methodsmentioning

confidence: 99%

Section: Meta-data and Global Clusteringmentioning

confidence: 99%

Learning Resolution Parameters for Graph Clustering

Veldt

Gleich

Wirth

2019

The World Wide Web Conference

Self Cite

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“…A series of work (Brickell et al 2008;Duggal et al 2013;Boytsov and Naidan 2013;Gilbert and Jain 2017;Fan, Raichek, and Van Buskirk 2018;Gilbert and Sonthalia 2018;Veldt et al 2018;Ruggles, Veldt, and Gleich 2020) models the scenario of repairing a metric under different settings, among which we are interested in a specific metric nearness model in this paper. For a set of input dissimilarity values between pairs of data samples, the model aims to output a set of distances that satisfy the metric constraints while keeping the output distances as near as possible to the input dissimilarity according to a certain measure of nearness.…”

Section: Introductionmentioning

confidence: 99%

Metric Nearness Made Practical

2023

AAAI

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Given a square matrix with noisy dissimilarity measures between pairs of data samples, the metric nearness model computes the best approximation of the matrix from a set of valid distance metrics. Despite its wide applications in machine learning and data processing tasks, the model faces non-trivial computational requirements in seeking the solution due to the large number of metric constraints associated with the feasible region. Our work designed a practical approach in two stages to tackle the challenge and improve the model's scalability and applicability. The first stage computes a fast yet high-quality approximate solution from a set of isometrically embeddable metrics, further improved by an effective heuristic. The second stage refines the approximate solution with the Halpern-Lions-Wittmann-Bauschke projection algorithm, which converges quickly to the optimal solution. In empirical evaluations, the proposed approach runs at least an order of magnitude faster than the state-of-the-art solutions, with significantly improved scalability, complete conformity to constraints, less memory consumption, and other desirable features in real applications.

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“…If a good LP lower bound can be computed, the output of fast heuristic methods can be compared against the LP lower bound to obtain a posteriori approximation guarantees that are often very good in practice [26,39,43,48]. However, despite some recent work on specialized solvers for these linear programs [33,37,44], these lower bounds can only be computed for medium-sized instances at best, and even then this can take a long time. There is therefore a need for faster approximation algorithms for correlation clustering, as well as an even more basic need to efficiently compute good lower bounds for the NP-hard objective in practice.…”

Section: Introductionmentioning

confidence: 99%

Correlation Clustering via Strong Triadic Closure Labeling: Fast Approximation Algorithms and Practical Lower Bounds

Veldt¹

2021

Preprint

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Correlation clustering is a framework for partitioning datasets based on pairwise similarity and dissimilarity scores, and has been used for diverse applications in bioinformatics, social network analysis, and computer vision. Although many approximation algorithms have been designed for this problem, the best theoretical results rely on obtaining lower bounds via expensive linear programming relaxations. In this paper we prove new relationships between correlation clustering problems and edge labeling problems related to the principle of strong triadic closure. We use these connections to develop new approximation algorithms for correlation clustering that have deterministic constant factor approximation guarantees and avoid the canonical linear programming relaxation. Our approach also extends to a variant of correlation clustering called cluster deletion, that strictly prohibits placing negative edges inside clusters. Our results include 4-approximation algorithms for cluster deletion and correlation clustering, based on simplified linear programs with far fewer constraints than the canonical relaxations. More importantly, we develop faster techniques that are purely combinatorial, based on computing maximal matchings in certain auxiliary graphs and hypergraphs. This leads to a combinatorial 6-approximation for complete unweighted correlation clustering, which is the best deterministic result for any method that does not rely on linear programming. We also present the first combinatorial constant factor approximation for cluster deletion.

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A Parallel Projection Method for Metric Constrained Optimization

Cited by 4 publications

References 27 publications

Learning Resolution Parameters for Graph Clustering

Learning Resolution Parameters for Graph Clustering

Metric Nearness Made Practical

Correlation Clustering via Strong Triadic Closure Labeling: Fast Approximation Algorithms and Practical Lower Bounds

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