Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps

TL;DR

This paper establishes near-optimal lower bounds of order n log n for the bounded coefficient complexity of polynomial multiplication and division, using advanced linear algebra and circuit complexity techniques.

Contribution

It introduces a novel approach to lower bounds in bounded coefficient circuits, extending Raz's ideas and unitarily invariant bounds to new bilinear problems.

Findings

Proves n log n lower bounds for polynomial multiplication and division.

Extends bounds to circuits with limited unbounded scalar multiplications.

Provides a new lower bound on the complexity of linear forms based on singular values.

Abstract

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.