Optimal Strong Parallel Repetition for Projection Games on Low Threshold Rank Graphs

Abstract

Given a two-player one-round game G with value val(G) = (1 − η), how quickly does the value decay under parallel repetition? If G is a projection game, then it is known that we can guarantee val(G ^⊗ n) ≤ (1 − η ²)^Ω(n), and that this is optimal. An important question is under what conditions can we guarantee that strong parallel repetition holds, i.e. val(G ^⊗) ≤ (1 − η)^Ω(n)?

In this work, we show a strong parallel repetition theorem for the case when G’s constraint graph has low threshold rank. In particular, for any k ≥ 2, if σ _k is the k-th largest singular value of G’s constraint graph, then we show that

$$ val(G^{\otimes n}) \leq \left(1 - \frac{\sqrt{1-\sigma_k^2}}{k}\cdot \eta\right)^{\Omega(n)}. $$

This improves and generalizes upon the work of [RR12], who showed a strong parallel repetition theorem for the case when G’s constraint graph is an expander.