Fuzzy Clustering with Similarity Queries

• URL: http://arxiv.org/abs/2106.02212v1
• Date: Fri, 4 Jun 2021 02:32:26 GMT
• Title: Fuzzy Clustering with Similarity Queries
• Authors: Wasim Huleihel, Arya Mazumdar, Soumyabrata Pal
• Abstract summary: The fuzzy or soft objective is a popular generalization of the well-known $k$-means problem. We show that by making few queries, the problem becomes easier to solve.
• Score: 56.96625809888241
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The fuzzy or soft $k$-means objective is a popular generalization of the well-known $k$-means problem, extending the clustering capability of the $k$-means to datasets that are uncertain, vague, and otherwise hard to cluster. In this paper, we propose a semi-supervised active clustering framework, where the learner is allowed to interact with an oracle (domain expert), asking for the similarity between a certain set of chosen items. We study the query and computational complexities of clustering in this framework. We prove that having a few of such similarity queries enables one to get a polynomial-time approximation algorithm to an otherwise conjecturally NP-hard problem. In particular, we provide probabilistic algorithms for fuzzy clustering in this setting that asks $O(\mathsf{poly}(k)\log n)$ similarity queries and run with polynomial-time-complexity, where $n$ is the number of items. The fuzzy $k$-means objective is nonconvex, with $k$-means as a special case, and is equivalent to some other generic nonconvex problem such as non-negative matrix factorization. The ubiquitous Lloyd-type algorithms (or, expectation-maximization algorithm) can get stuck at a local minima. Our results show that by making few similarity queries, the problem becomes easier to solve. Finally, we test our algorithms over real-world datasets, showing their effectiveness in real-world applications.

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