Capacity-Resolution Trade-Off in the Optimal Learning of Multiple Low-Dimensional Manifolds by Attractor Neural Networks.

Recurrent neural networks (RNN) are powerful tools to explain how attractors may emerge from noisy, high-dimensional dynamics. We study here how to learn the ∼N^{2} pairwise interactions in a RNN with N neurons to embed L manifolds of dimension D≪N. We show that the capacity, i.e., the maximal ratio L/N, decreases as |logε|^{-D}, where ε is the error on the position encoded by the neural activity along each manifold. Hence, RNN are flexible memory devices capable of storing a large number of manifolds at high spatial resolution. Our results rely on a combination of analytical tools from statistical mechanics and random matrix theory, extending Gardner's classical theory of learning to the case of patterns with strong spatial correlations.