Sign-Rank, Index, and List Replicability: Connections and Separations

2026-06-16

AI Summary

This work addresses the longstanding challenge of constructing lower bounds for sign rank by establishing a novel connection between the ℤ₂-index and list replicability. We prove, for the first time, that the ℤ₂-index is linearly upper-bounded by the list replicability number, thereby achieving a strong separation between sign rank and the ℤ₂-index. This insight yields new combinatorial upper bounds and a composition lemma for list replicability. By integrating tools from combinatorics, topological methods, and representation complexity, we resolve an open problem posed by Frick et al., uncovering intrinsic relationships between list replicability and combinatorial parameters such as height and minimum star number. Furthermore, we provide an upper bound on the list replicability of product concept classes.

Abstract

In learning theory, the sign rank of a binary concept class captures the smallest dimension in which it can be represented by points and halfspaces. Despite tremendous interest, lower bounds on sign rank are notoriously difficult to come by. Two recent approaches to the problem establish lower bounds on sign rank by measures that are easier to analyze: the $\mathbb{Z}_2$-index and the list replicability number. We order these measures, showing that the $\mathbb{Z}_2$-index is upper-bounded by a linear function of the list replicability number. As a main consequence, we obtain a strong separation between sign rank and $\mathbb{Z}_2$-index, thereby resolving a question of Frick, Hosseini, and Vasileuski. This motivates a thorough study of list replicability, the stronger of the two lower-bounding measures. We establish upper bounds on the list replicability number by two combinatorial measures: height and minimum star number. We also prove a fundamental composition result, showing that the product of two concept classes has list replicability number bounded by the sum of the list replicability numbers of the two classes.

Problem

• sign rank
• Z2-index
• list replicability
• learning theory
• lower bounds

Innovation

• sign rank
• list replicability
• ℤ₂-index
• combinatorial bounds
• concept class composition