A Bivariate Invariance Principle

TL;DR

This paper extends the Basic Invariance Principle to bivariate multilinear polynomials, providing a tighter analysis for separable functions and deriving a version for polynomials with random coefficients.

Contribution

It introduces a bivariate invariance principle, generalizing the BIP, and demonstrates its advantages over naive approaches for certain function classes.

Findings

The bivariate invariance principle is exponentially tighter for separable functions.

A version of BIP for polynomials with random coefficients is derived.

The naive bivariate approach is less effective for specific function classes.

Abstract

A notable result from analysis of Boolean functions is the Basic Invariance Principle (BIP), a quantitative nonlinear generalization of the Central Limit Theorem for multilinear polynomials. We present a generalization of the BIP for bivariate multilinear polynomials, i.e., polynomials over two n-length sequences of random variables. This bivariate invariance principle arises from an iterative application of the BIP to bound the error in replacing each of the two input sequences. In order to prove this invariance principle, we first derive a version of the BIP for random multilinear polynomials, i.e., polynomials whose coefficients are random variables. As a benchmark, we also state a naive bivariate invariance principle which treats the two input sequences as one and directly applies the BIP. Neither principle is universally stronger than the other, but we do show that for a notable class of bivariate functions, which we term separable functions, our subtler principle is exponentially tighter than the naive benchmark.