### 最新论文

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 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 alternating minimization algorithms) can get stuck at a local minimum. Our results show that by making a 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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