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

Thibault Lesieur (1) , Caterina de Bacco (2) , Jess Banks (2) , Florent Krzakala (3) , Cris Moore (2) , Lenka Zdeborová (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 Santa Fe Institute
3 LPS - Laboratoire de Physique Statistique de l'ENS

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.

Keywords: Cond-mat; Cond-mat.dis-nn; Cs; Cs.IT; Math; Stat

Domains: Other Statistics [stat.ML]; Information Theory [math.IT]

Thibault Lesieur, Caterina de Bacco, Jess Banks, Florent Krzakala, Cris Moore, et al.. Phase transitions and optimal algorithms in high-dimensional Gaussian mixture clustering. 2016. ⟨cea-01448112⟩