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Register a new userHierarchical Clustering with Structural Constraints
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Rad Niazadeh
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Hierarchical clustering is a popular unsupervised data analysis method. For many real-world applications, we would like to exploit prior information about the data that imposes constraints on the clustering hierarchy, and is not captured by the set of features available to the algorithm. This gives rise to the problem of hierarchical clustering with structural constraints. Structural constraints pose major challenges for bottom-up approaches like average/single linkage and even though they can be naturally incorporated into top-down divisive algorithms, no formal guarantees exist on the quality of their output. In this paper, we provide provable approximation guarantees for two simple top-down algorithms, using a recently introduced optimization viewpoint of hierarchical clustering with pairwise similarity information [Dasgupta, 2016]. We show how to find good solutions even in the presence of conflicting prior information, by formulating a constraint-based regularization of the objective. We further explore a variation of this objective for dissimilarity information [Cohen-Addad et al., 2018] and improve upon current techniques. Finally, we demonstrate our approach on a real dataset for the taxonomy application.
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Hierarchical Clustering (HC) is a widely studied problem in exploratory data analysis, usually tackled by simple agglomerative procedures like average-linkage, single-linkage or complete-linkage. In this paper we focus on two objectives, introduced recently to give insight into the performance of average-linkage clustering: a similarity based HC objective proposed by [Moseley and Wang, 2017] and a dissimilarity based HC objective proposed by [Cohen-Addad et al., 2018]. In both cases, we present tight counterexamples showing that average-linkage cannot obtain better than 1/3 and 2/3 approximations respectively (in the worst-case), settling an open question raised in [Moseley and Wang, 2017]. This matches the approximation ratio of a random solution, raising a natural question: can we beat average-linkage for these objectives? We answer this in the affirmative, giving two new algorithms based on semidefinite programming with provably better guarantees.
Recently, Hierarchical Clustering (HC) has been considered through the lens of optimization. In particular, two maximization objectives have been defined. Moseley and Wang defined the emph{Revenue} objective to handle similarity information given by a weighted graph on the data points (w.l.o.g., $[0,1]$ weights), while Cohen-Addad et al. defined the emph{Dissimilarity} objective to handle dissimilarity information. In this paper, we prove structural lemmas for both objectives allowing us to convert any HC tree to a tree with constant number of internal nodes while incurring an arbitrarily small loss in each objective. Although the best-known approximations are 0.585 and 0.667 respectively, using our lemmas we obtain approximations arbitrarily close to 1, if not all weights are small (i.e., there exist constants $epsilon, delta$ such that the fraction of weights smaller than $delta$, is at most $1 - epsilon$); such instances encompass many metric-based similarity instances, thereby improving upon prior work. Finally, we introduce Hierarchical Correlation Clustering (HCC) to handle instances that contain similarity and dissimilarity information simultaneously. For HCC, we provide an approximation of 0.4767 and for complementary similarity/dissimilarity weights (analogous to $+/-$ correlation clustering), we again present nearly-optimal approximations.
Recent works on Hierarchical Clustering (HC), a well-studied problem in exploratory data analysis, have focused on optimizing various objective functions for this problem under arbitrary similarity measures. In this paper we take the first step and give novel scalable algorithms for this problem tailored to Euclidean data in R^d and under vector-based similarity measures, a prevalent model in several typical machine learning applications. We focus primarily on the popular Gaussian kernel and other related measures, presenting our results through the lens of the objective introduced recently by Moseley and Wang [2017]. We show that the approximation factor in Moseley and Wang [2017] can be improved for Euclidean data. We further demonstrate both theoretically and experimentally that our algorithms scale to very high dimension d, while outperforming average-linkage and showing competitive results against other less scalable approaches.
In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a_1,ldots,a_n} over Sigma^m, where Sigma is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. That is, we seek a partition {I_1,ldots,I_k} of {1,ldots,n} and vectors c_1,ldots,c_kinSigma^m such that sum_{i=1}^ksum_{jin I_i}d_h(c_i,a_j)leq B, where d_H(a,b) is the Hamming distance between vectors a and b. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable (for binary case Sigma={0,1}) by giving an algorithm that solves the problem in time 2^{O(Blog B)} (mn)^{O(1)}. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters should satisfy certain constraints. More precisely, in Capacitated Clustering, in addition, we are given two non-negative integers p and q, and seek a clustering with pleq |I_i|leq q for all iin{1,ldots,k}. Our main theorem is that Capacitated Clustering is solvable in time 2^{O(Blog B)}|Sigma|^B(mn)^{O(1)}. The theorem not only extends the previous algorithmic results to a significantly more general model, it also implies algorithms for several other variants of Categorical Clustering with constraints on cluster sizes.
We study a recent inferential framework, named posterior regularisation, on the Bayesian hierarchical mixture clustering (BHMC) model. This framework facilitates a simple way to impose extra constraints on a Bayesian model to overcome some weakness of the original model. It narrows the search space of the parameters of the Bayesian model through a formalism that imposes certain constraints on the features of the found solutions. In this paper, in order to enhance the separation of clusters, we apply posterior regularisation to impose max-margin constraints on the nodes at every level of the hierarchy. This paper shows how the framework integrates with BHMC and achieves the expected improvements over the original Bayesian model.
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