Sign-Rank Can Increase Under Intersection

The communication class $\mathbf{UPP}^{\text{cc}}$ is a communication analog of the Turing Machine complexity class $\mathbf{PP}$. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem $f$, let $f \wedge f$ denote the function that evaluates $f$ on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem $f$ with $\mathbf{UPP}(f)= O(\log n)$, and $\mathbf{UPP}(f \wedge f) = \Theta(\log^2 n)$. This is the first result showing that $\mathbf{UPP}$ communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that $\mathbf{UPP}^{\text{cc}}$, the class of problems with polylogarithmic-cost $\mathbf{UPP}$ communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on $n$ bits has dimension complexity $n^{\Omega(\log n)}$. This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.