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


We consider the problem of Gaussian mixture clustering in the high-dimensional limit where the data consists of $m$ points in $n$ dimensions, $n,m\rightarrow\infty$ 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.

Proc. 54th Annual Allerton Conference on Communication, Control, and Computing (2016)
Caterina De Bacco
Caterina De Bacco
CyberValley Research Group Leader

My research focuses on understanding, optimizing and predicting relations between the microscopic and macroscopic properties of complex large-scale interacting systems.