2022/06/14 by Weishun Zhong, Ben Sorscher, Zhong, Weishun +5 · 1 citation
Computer Science · Neuroscience · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Biological sciences #FOS: Computer and information sciences #FOS: Physical sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Neural Networks and Applications #Neural dynamics and brain function #Neurons and Cognition (q-bio.NC) #Statistical Mechanics (cond-mat.stat-mech) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2206.08933
openalex publication_date 2022/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A central question in computational neuroscience is how structure determines function in neural networks. The emerging high-quality large-scale connectomic datasets raise the question of what general functional principles can be gleaned from structural information such as the distribution of excitatory/inhibitory synapse types and the distribution of synaptic weights. Motivated by this question, we developed a statistical mechanical theory of learning in neural networks that incorporates structural information as constraints. We derived an analytical solution for the memory capacity of the perceptron, a basic feedforward model of supervised learning, with constraint on the distribution of its weights. Our theory predicts that the reduction in capacity due to the constrained weight-distribution is related to the Wasserstein distance between the imposed distribution and that of the standard normal distribution. To test the theoretical predictions, we use optimal transport theory and information geometry to develop an SGD-based algorithm to find weights that simultaneously learn the input-output task and satisfy the distribution constraint. We show that training in our algorithm can be interpreted as geodesic flows in the Wasserstein space of probability distributions. We further developed a statistical mechanical theory for teacher-student perceptron rule learning and ask for the best way for the student to incorporate prior knowledge of the rule. Our theory shows that it is beneficial for the learner to adopt different prior weight distributions during learning, and shows that distribution-constrained learning outperforms unconstrained and sign-constrained learning. Our theory and algorithm provide novel strategies for incorporating prior knowledge about weights into learning, and reveal a powerful connection between structure and function in neural networks.