On the Ising perceptron model

(This lecture will be self-contained.) In high dimensions, what does it look like when we take the intersection of a set of random half-spaces with either the sphere or the Hamming cube? This is one phrasing of the so-called perceptron problem, whose study originated with a toy model of a very simple neural network. A non-rigorous solution was given in the 1980s (Gardner, Derrida, Krauth, Mézard) using methods of statistical physics. Shcherbina and Tirozzi gave a rigorous solution to the positive spherical perceptron, making crucial use of the convexity of this problem. The other cases of the model are non-convex and remain open problems. I will survey the physics solution and describe some results for the perceptron on the Hamming cube.

This lecture is based on joint work with Jian Ding.



Massachusetts Institute of Technology


Nike Sun