Phase transitions in optimal unsupervised learning

We determine the optimal performance of learning the orientation of the symmetry axis of a set of $P=\alpha N$ points that are uniformly distributed in all the directions but one on the $N$-dimensional space. The components along the symmetry breaking direction, of unitary vector $\mathbf{B}$, are sampled from a mixture of two Gaussians of variable separation and width. The typical optimal performance is measured through the overlap $R_{\mathrm{opt}}=\mathbf{B}\cdot {\mathbf{J}}^{*}$, where ${\mathbf{J}}^{*}$ is the optimal guess of the symmetry breaking direction. Within this general scenario, the learning curves $R_{\mathrm{opt}}\left(\alpha \right)$ may present first order transitions if the clusters are narrow enough. Close to these transitions, high performance states can be obtained through the minimization of the corresponding optimal potential, although these solutions are metastable, and therefore not learnable, within the usual Bayesian scenario.