Cortical Computation via Iterative Constructions

TL;DR

This paper investigates simple iterative constructions for Boolean functions, demonstrating their ability to approximate threshold functions efficiently and exploring their convergence, learnability, and biological plausibility.

Contribution

It generalizes Valiant's majority construction to all uniform threshold functions using primitive-based iterative methods, analyzing convergence rates and learnability.

Findings

Linear convergence achievable with fixed primitives

Quadratic convergence requires growing primitive size near thresholds

Errors increase near threshold boundaries, matching cognitive observations

Abstract

We study Boolean functions of an arbitrary number of input variables that can be realized by simple iterative constructions based on constant-size primitives. This restricted type of construction needs little global coordination or control and thus is a candidate for neurally feasible computation. Valiant's construction of a majority function can be realized in this manner and, as we show, can be generalized to any uniform threshold function. We study the rate of convergence, finding that while linear convergence to the correct function can be achieved for any threshold using a fixed set of primitives, for quadratic convergence, the size of the primitives must grow as the threshold approaches 0 or 1. We also study finite realizations of this process and the learnability of the functions realized. We show that the constructions realized are accurate outside a small interval near the target threshold, where the size of the construction grows as the inverse square of the interval width. This phenomenon, that errors are higher closer to thresholds (and thresholds closer to the boundary are harder to represent), is a well-known cognitive finding.