Clustering with Same-Cluster Queries
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We propose a framework for Semi-Supervised Active Clustering framework (SSAC), where the learner is allowed to interact with a domain expert, asking whether two given instances belong to the same cluster or not. We study the query and computational complexity of clustering in this framework. We consider a setting where the expert conforms to a center-based clustering with a notion of margin. We show that there is a trade off between computational complexity and query complexity; We prove that for the case of k-means clustering (i.e., when the expert conforms to a solution of k-means), having access to relatively few such queries allows efficient solutions to otherwise NP hard problems. In particular, we provide a probabilistic polynomial-time (BPP) algorithm for clustering in this setting that asks O(k^2 k + k n) same-cluster queries and runs with time complexity O(kn n) (where k is the number of clusters and n is the number of instances). The algorithm succeeds with high probability for data satisfying margin conditions under which, without queries, we show that the problem is NP hard. We also prove a lower bound on the number of queries needed to have a computationally efficient clustering algorithm in this setting.
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