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Abstract In recent years, significant attention in deep learning theory has been devoted to analyzing when models that interpolate their training data can still generalize well to unseen examples. Many insights have been gained from studying models with multiple layers of Gaussian random features, for which one can compute precise generalization asymptotics. However, few works have considered the effect of weight anisotropy; most assume that the random features are generated using independent and identically distributed Gaussian weights, and allow only for structure in the input data. Here, we use the replica trick from statistical physics to derive learning curves for models with many layers of structured Gaussian features. We show that allowing correlations between the rows of the first layer of features can aid generalization, while structure in later layers is generally detrimental. Our results shed light on how weight structure affects generalization in a simple class of solvable models. 

Abstract Sequence memory is an essential attribute of natural and artificial intelligence that enables agents to encode, store, and retrieve complex sequences of stimuli and actions. Computational models of sequence memory have been proposed where recurrent Hopfield-like neural networks are trained with temporally asymmetric Hebbian rules. However, these networks suffer from limited sequence capacity (maximal length of the stored sequence) due to interference between the memories. Inspired by recent work on Dense Associative Memories, we expand the sequence capacity of these models by introducing a nonlinear interaction term, enhancing separation between the patterns. We derive novel scaling laws for sequence capacity with respect to network size, significantly outperforming existing scaling laws for models based on traditional Hopfield networks, and verify these theoretical results with numerical simulation. Moreover, we introduce a generalized pseudoinverse rule to recall sequences of highly correlated patterns. Finally, we extend this model to store sequences with variable timing between states’ transitions and describe a biologically-plausible implementation, with connections to motor neuroscience.