Exact versus Approximate Representations of Boolean Functions in the De Morgan Basis

Abstract

A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the function, are at most polynomially separated. Extending this result from degree to other complexity measures like sparsity of the polynomial representation, or total weight of the coefficients, remains poorly understood. In this work, we consider this problem in the De Morgan basis, and prove an analogous result for the sparsity of the polynomials at a logarithmic scale. Our result further implies that the exact $\ell_1$ norm and its approximate variant are also similarly related to each other at a log scale. This is in contrast to the Fourier basis, where the analog of our results are known to be false. Our proof is based on a novel random restriction method. Unlike most existing random restriction methods used in complexity theory, our random restriction process is adaptive and is based on how various complexity measures simplify during the restriction process.