On the tightness of an SDP relaxation of k-means
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Recently, Awasthi et al. introduced an SDP relaxation of the k-means problem in R^m. In this work, we consider a random model for the data points in which k balls of unit radius are deterministically distributed throughout R^m, and then in each ball, n points are drawn according to a common rotationally invariant probability distribution. For any fixed ball configuration and probability distribution, we prove that the SDP relaxation of the k-means problem exactly recovers these planted clusters with probability 1-e^-Ω(n) provided the distance between any two of the ball centers is >2+ϵ, where ϵ is an explicit function of the configuration of the ball centers, and can be arbitrarily small when m is large.
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