  To improve the storage capacity of the Hopfield model, we develop a version
of the dreaming algorithm that perpetually reinforces the patterns to be stored
(as in the Hebb rule), and erases the spurious memories (as in dreaming
algorithms). For this reason, we called it Daydreaming. Daydreaming is not
destructive and it converges asymptotically to stationary retrieval maps. When
trained on random uncorrelated examples, the model shows optimal performance in
terms of the size of the basins of attraction of stored examples and the
quality of reconstruction. We also train the Daydreaming algorithm on
correlated data obtained via the random-features model and argue that it
spontaneously exploits the correlations thus increasing even further the
storage capacity and the size of the basins of attraction. Moreover, the
Daydreaming algorithm is also able to stabilize the features hidden in the
data. Finally, we test Daydreaming on the MNIST dataset and show that it still
works surprisingly well, producing attractors that are close to unseen examples
and class prototypes.
