In search of dispersed memories: Generative diffusion models are associative memory networks

Hopfield networks are widely used in neuroscience as simplified theoretical models of biological associative memory. The original Hopfield networks store memories by encoding patterns of binary associations, which result in a synaptic learning mechanism known as Hebbian learning rule. Modern Hopfield networks can achieve exponential capacity scaling by using highly non-linear energy functions. However, the energy function of these newer models cannot be straightforwardly compressed into binary synaptic couplings and it does not directly provide new synaptic learning rules. In this work we show that generative diffusion models can be interpreted as energy-based models and that, when trained on discrete patterns, their energy function is equivalent to that of modern Hopfield networks. This equivalence allows us to interpret the supervised training of diffusion models as a synaptic learning process that encodes the associative dynamics of a modern Hopfield network in the weight structure of a deep neural network. Accordingly, in our experiments we show that the storage capacity of a continuous modern Hopfield network is identical to the capacity of a diffusion model. Our results establish a strong link between generative modeling and the theoretical neuroscience of memory, which provide a powerful computational foundation for the reconstructive theory of memory, where creative generation and memory recall can be seen as parts of a unified continuum.