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Register a new userApproximation Algorithms for Bregman Co-clustering and Tensor Clustering
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Stefanie Jegelka
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor clustering [8,34]. Like k-means, these more general problems also suffer from the NP-hardness of the associated optimization. Researchers have developed approximation algorithms of varying degrees of sophistication for k-means, k-medians, and more recently also for Bregman clustering [2]. However, there seem to be no approximation algorithms for Bregman co- and tensor clustering. In this paper we derive the first (to our knowledge) guaranteed methods for these increasingly important clustering settings. Going beyond Bregman divergences, we also prove an approximation factor for tensor clustering with arbitrary separable metrics. Through extensive experiments we evaluate the characteristics of our method, and show that it also has practical impact.
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We present an $(e^{O(p)} frac{log ell}{loglogell})$-approximation algorithm for socially fair clustering with the $ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $sum_{u text{ in group }j} d(u,C)^p$. The socially fair clustering problem was independently proposed by Ghadiri, Samadi, and Vempala [2021] and Abbasi, Bhaskara, and Venkatasubramanian [2021]. Our algorithm improves and generalizes their $O(ell)$-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $Omega(ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $Theta(frac{log ell}{loglogell})$ for a fixed $p$. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. [2021].
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Hierarchical clustering is a fundamental task often used to discover meaningful structures in data, such as phylogenetic trees, taxonomies of concepts, subtypes of cancer, and cascades of particle decays in particle physics. Typically approximate algorithms are used for inference due to the combinatorial number of possible hierarchical clusterings. In contrast to existing methods, we present novel dynamic-programming algorithms for emph{exact} inference in hierarchical clustering based on a novel trellis data structure, and we prove that we can exactly compute the partition function, maximum likelihood hierarchy, and marginal probabilities of sub-hierarchies and clusters. Our algorithms scale in time and space proportional to the powerset of $N$ elements which is super-exponentially more efficient than explicitly considering each of the (2N-3)!! possible hierarchies. Also, for larger datasets where our exact algorithms become infeasible, we introduce an approximate algorithm based on a sparse trellis that compares well to other benchmarks. Exact methods are relevant to data analyses in particle physics and for finding correlations among gene expression in cancer genomics, and we give examples in both areas, where our algorithms outperform greedy and beam search baselines. In addition, we consider Dasguptas cost with synthetic data.
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