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

Given a two-player one-round game G with value val(G) = (1-eta), 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(circle times n)) <= (1 -eta(2))(Omega(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(circle times)) <= (1 - eta)(Omega(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 sigma(k) is the k-th largest singular value of G's constraint graph, then we show that val(G(circle times n)) <= (1 - root 1-sigma(2)(k) / k .eta)(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.