Self averaging and the space of interactions in neural networks

Abstract

We prove through a precise exponential inequality that the logarithm of the N Ž size of the intersection of M random half spaces with the unit sphere of ‫ޒ‬ resp., the Ä 4 N . discrete cube y1, 1 is, as N ª ϱ, a self averaging quantity. This provides justification for w Ž .x one of the first steps of a famous computation by E. Gardner

Sharp threshold sequence and universality for Ising perceptron models

“…It would be of interest to see if the methods of [Xu21] can be extended to more general perceptron models, in ways that do not require more precise estimates on these models [Xu22]. With respect to our current paper, the most closely related previous results are the estimates obtained by Talagrand for the half-space perceptron model [Tal99b,Tal11].…”

Gardner formula for Ising perceptron models at small densities

“…A special case of this result, for the half-space model p q " 1t ě u, was previously obtained in [Tal11a, Ch. 9] (with partial results appearing in a previous work [Tal99b]). 5 Talagrand's proof for the half-space model relies crucially on an estimate [Tal11b,m.…”

Pattern capacity of a single quantum perceptron

“…The activation function is then evaluated on the measured value of the transition probability |∆ µ | 2 . Instead, in our case we infer the value of sgn(∆ µ ) by measuring a gaussianly distributed parameter (see (20) and the corresponding discussion), which mimics the functioning of a stochastic classical perceptron. Using the same statistical techniques employed here, the authors find that the pattern capacity of that model is twice the classical capacity.…”