On Algebraic Constructions of Neural Networks with Small Weights

TL;DR

This paper introduces an algebraic framework to construct neural networks with small weights, improving the understanding of complexity trade-offs and providing explicit constructions for functions like EQUALITY and COMPARISON.

Contribution

It develops new algebraic techniques to reduce weight sizes in neural networks, including explicit constructions for key functions, advancing circuit complexity theory.

Findings

Explicit constant weight matrix for EQUALITY function

Existence of linear weight matrices for COMPARISON function

Improved upper bounds on weight sizes for neural circuit functions

Abstract

Neural gates compute functions based on weighted sums of the input variables. The expressive power of neural gates (number of distinct functions it can compute) depends on the weight sizes and, in general, large weights (exponential in the number of inputs) are required. Studying the trade-offs among the weight sizes, circuit size and depth is a well-studied topic both in circuit complexity theory and the practice of neural computation. We propose a new approach for studying these complexity trade-offs by considering a related algebraic framework. Specifically, given a single linear equation with arbitrary coefficients, we would like to express it using a system of linear equations with smaller (even constant) coefficients. The techniques we developed are based on Siegel's Lemma for the bounds, anti-concentration inequalities for the existential results and extensions of Sylvester-type Hadamard matrices for the constructions. We explicitly construct a constant weight, optimal size matrix to compute the EQUALITY function (checking if two integers expressed in binary are equal). Computing EQUALITY with a single linear equation requires exponentially large weights. In addition, we prove the existence of the best-known weight size (linear) matrices to compute the COMPARISON function (comparing between two integers expressed in binary). In the context of the circuit complexity theory, our results improve the upper bounds on the weight sizes for the best-known circuit sizes for EQUALITY and COMPARISON.