Sign-Rank, Index, and List Replicability: Connections and Separations
For learning theorists, the paper provides new lower-bound techniques for sign rank and resolves a known open problem, though the results are incremental in nature.
The paper orders the Z2-index and list replicability number, showing the former is bounded by a linear function of the latter, and uses this to prove a strong separation between sign rank and Z2-index, resolving an open question. It also establishes upper bounds on list replicability by height and minimum star number, and proves a composition result for product classes.
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.