# Dual polynomials and communication complexity of $\textsf{XOR}$ functions

## TL;DR

This paper establishes a duality between polynomial margin complexity and discrepancy for XOR functions, leading to new bounds and characterizations of communication complexity, spectral norms, and circuit size for symmetric functions.

## Contribution

It introduces a duality framework connecting polynomial margin complexity and discrepancy, and applies it to derive bounds on communication complexity and circuit size for XOR functions.

## Key findings

- PP complexity of f XOR is Omega(k / log(n/k))
- Exponential separation between PP^cc and UPP^cc via XOR functions
- Characterization of approximate spectral norm of symmetric functions

## Abstract

We show a new duality between the polynomial margin complexity of $f$ and the discrepancy of the function $f \circ \textsf{XOR}$, called an $\textsf{XOR}$ function. Using this duality, we develop polynomial based techniques for understanding the bounded error ($\textsf{BPP}$) and the weakly-unbounded error ($\textsf{PP}$) communication complexities of $\textsf{XOR}$ functions. We show the following. A weak form of an interesting conjecture of Zhang and Shi (Quantum Information and Computation, 2009) (The full conjecture has just been reported to be independently settled by Hatami and Qian (Arxiv, 2017). However, their techniques are quite different and are not known to yield many of the results we obtain here). Zhang and Shi assert that for symmetric functions $f : \{0, 1\}^n \rightarrow \{-1, 1\}$, the weakly unbounded-error complexity of $f \circ \textsf{XOR}$ is essentially characterized by the number of points $i$ in the set $\{0,1, \dots,n-2\}$ for which $D_f(i) \neq D_f(i+2)$, where $D_f$ is the predicate corresponding to $f$. The number of such points is called the odd-even degree of $f$. We show that the $\textsf{PP}$ complexity of $f \circ \textsf{XOR}$ is $\Omega(k/ \log(n/k))$. We resolve a conjecture of a different Zhang characterizing the Threshold of Parity circuit size of symmetric functions in terms of their odd-even degree. We obtain a new proof of the exponential separation between $\textsf{PP}^{cc}$ and $\textsf{UPP}^{cc}$ via an $\textsf{XOR}$ function. We provide a characterization of the approximate spectral norm of symmetric functions, affirming a conjecture of Ada et al. (APPROX-RANDOM, 2012) which has several consequences. Additionally, we prove strong $\textsf{UPP}$ lower bounds for $f \circ \textsf{XOR}$, when $f$ is symmetric and periodic with period $O(n^{1/2-\epsilon})$, for any constant $\epsilon > 0$.