Lifting with Inner Functions of Polynomial Discrepancy
Yahel Manor, Or Meir
TL;DR
The paper proves a new lifting theorem for f∘g^n in which the inner function g has small discrepancy, quantified as Δ(g) ≥ c log n, extending previous work to a broader class of inner functions. Central to the approach is a simulation framework that converts any protocol for f∘g^n into a decision-tree (deterministic or randomized) solving f with query complexity proportional to the outer complexity divided by Δ(g). A key technical advance is the introduction of recoverable values and density-restoring mechanisms, enabling the main lemma to hold even when Δ(g) is not as large as in prior results, and ensuring the density and entropy conditions are preserved during the simulation. The results include both deterministic and randomized lifting theorems, as well as a discussion of tightness and extensions to product-distribution discrepancies, highlighting broader implications for direct-sum, XOR-type, and data-structural lower bound contexts. Overall, the work broadens the applicability of lifting theorems and deepens the connection between discrepancy, information cost, and the complexity of composed problems.
Abstract
Lifting theorems are theorems that bound the communication complexity of a composed function $f\circ g^{n}$ in terms of the query complexity of $f$ and the communication complexity of $g$. Such theorems constitute a powerful generalization of direct-sum theorems for $g$, and have seen numerous applications in recent years. We prove a new lifting theorem that works for every two functions $f,g$ such that the discrepancy of $g$ is at most inverse polynomial in the input length of $f$. Our result is a significant generalization of the known direct-sum theorem for discrepancy, and extends the range of inner functions $g$ for which lifting theorems hold.