Sign-Rank of $k$-Hamming Distance is Constant

TL;DR

This paper proves that the sign-rank of the $k$-Hamming Distance matrix is bounded by a constant depending only on $k$, refuting previous conjectures that it depends on the number of bits $n$, with implications for communication complexity.

Contribution

It establishes a constant upper bound on the sign-rank of $k$-Hamming Distance matrices, challenging prior beliefs about their dependence on $n$, and introduces techniques applicable to related matrices.

Findings

Sign-rank of $k$-Hamming Distance matrix is $2^{O(k)}$

Refutes the conjecture that sign-rank depends on $n$

Provides bounds for related large-margin matrices

Abstract

We prove that the sign-rank of the $k$-Hamming Distance matrix on $n$ bits is $2^{O(k)}$, independent of the number of bits $n$. This strongly refutes the conjecture of Hatami, Hatami, Pires, Tao, and Zhao (RANDOM 2022), and Hatami, Hosseini, and Meng (STOC 2023), repeated in several other papers, that the sign-rank should depend on $n$. This conjecture would have qualitatively separated margin from sign-rank (or, equivalently, bounded-error from unbounded-error randomized communication). In fact, our technique gives constant sign-rank upper bounds for all matrices which reduce to $k$-Hamming Distance, as well as large-margin matrices recently shown to be irreducible to $k$-Hamming Distance.