Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering
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We consider the problem of Gaussian mixture clustering in the high-dimensional limit where the data consists of m points in n dimensions, n,m →∞ and α = m/n stays finite. Using exact but non-rigorous methods from statistical physics, we determine the critical value of α and the distance between the clusters at which it becomes information-theoretically possible to reconstruct the membership into clusters better than chance. We also determine the accuracy achievable by the Bayes-optimal estimation algorithm. In particular, we find that when the number of clusters is sufficiently large, r > 4 + 2 √(α), there is a gap between the threshold for information-theoretically optimal performance and the threshold at which known algorithms succeed.
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