Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering

Abstract

We consider the problem of Gaussian mixture clustering in the high-dimensional limit where the data consists of $m$ points in $n$ dimensions, $n,m\to \infty$ and $\alpha =m/n$ stays finite. Using exact but non-rigorous methods from statistical physics, we determine the critical value of $\alpha$ 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, $r>4+2\sqrt{\alpha }$, there is a gap between the threshold for information-theoretically optimal performance and the threshold at which known algorithms succeed.

8 pages, 3 figures, conference

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