Daniel D Lee

Weishun Zhong, Ben Sorscher, Daniel D Lee et al.

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.

Eric Mitchell, Selim Engin, Volkan Isler et al.

We present a new approach to 3D object representation where a neural network encodes the geometry of an object directly into the weights and biases of a second 'mapping' network. This mapping network can be used to reconstruct an object by applying its encoded transformation to points randomly sampled from a simple geometric space, such as the unit sphere. We study the effectiveness of our method through various experiments on subsets of the ShapeNet dataset. We find that the proposed approach can reconstruct encoded objects with accuracy equal to or exceeding state-of-the-art methods with orders of magnitude fewer parameters. Our smallest mapping network has only about 7000 parameters and shows reconstruction quality on par with state-of-the-art object decoder architectures with millions of parameters. Further experiments on feature mixing through the composition of learned functions show that the encoding captures a meaningful subspace of objects.