Storage and Learning phase transitions in the Random-Features Hopfield Model
This work addresses a theoretical problem in neural network modeling for researchers in statistical physics and machine learning, representing an incremental extension of the Hopfield model.
The authors tackled the problem of extending the Hopfield model to incorporate random features inspired by the manifold hypothesis, and derived its phase diagram using the replica method, uncovering a new 'learning phase' where features can be inferred from patterns.
The Hopfield model is a paradigmatic model of neural networks that has been analyzed for many decades in the statistical physics, neuroscience, and machine learning communities. Inspired by the manifold hypothesis in machine learning, we propose and investigate a generalization of the standard setting that we name Random-Features Hopfield Model. Here $P$ binary patterns of length $N$ are generated by applying to Gaussian vectors sampled in a latent space of dimension $D$ a random projection followed by a non-linearity. Using the replica method from statistical physics, we derive the phase diagram of the model in the limit $P,N,D\to\infty$ with fixed ratios $α=P/N$ and $α_D=D/N$. Besides the usual retrieval phase, where the patterns can be dynamically recovered from some initial corruption, we uncover a new phase where the features characterizing the projection can be recovered instead. We call this phenomena the learning phase transition, as the features are not explicitly given to the model but rather are inferred from the patterns in an unsupervised fashion.