PHASE-TRANSITIONS IN OPTIMAL UNSUPERVISED LEARNING

We determine the optimal performance of learning the orientation of th e symmetry axis of a set of P = alpha N points that are uniformly dist ributed in all the directions but one on the N-dimensional space. The components along the symmetry breaking direction, of unitary vector B, are sampled from a mixture of two Gaussians of variable separation an d width. The typical optimal performance is measured through the overl ap R-opt = B.J, where J* is the optimal guess of the symmetry breakin g direction. Within this general scenario, the learning curves R-opt(a lpha) may present first order transitions if the clusters an narrow en ough. Close to these transitions. high performance states can be obtai ned through the minimization of the corresponding optimal potential, a lthough these solutions are metastable, and therefore not learnable, w ithin the usual Bayesian scenario.