UPP$^{\text{cc}}$ is Not Closed Under Intersection
Mark Bun, Mandar Juvekar
The sign rank of a ${\pm 1}$-valued matrix $A$ is the minimum rank of a real matrix that agrees with $A$ entrywise in sign. For every sufficiently large $d$, we give an explicit matrix $A$ with $2^{Θ(d^2)}$ rows and columns and sign rank $Θ(d)$ such that $A \land A$ has sign rank $2^{Θ(d)}$. Here, $A \land A$ is the communication matrix of the problem obtained by evaluating $A$ on disjoint inputs and taking the conjunction of the results. As the logarithm of sign rank characterizes unbounded-error communication complexity, this implies that the communication class $\mathsf{UPP}^{\mathsf{cc}}$ is not closed under intersection. The starting point for our lower bound is Forster's method, which relates the sign rank of a matrix to the average margin of representations of its rows and columns by points and halfspaces. We study a continuous analog for conjunctions of inner-product sign tests on the unit sphere. Rotational symmetry connects this problem to polynomial sign representations, yielding an exponentially small upper bound on the average margin. This bound holds even when sign constraints are relaxed near the sign tests' decision boundary, allowing us to transfer it to explicit finite sign matrices arising from intersections of integer halfspaces.
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