On the optimality of truncation among binary linear sketches

We study statistical inference and function evaluation problems in which the input binary vector is available only through a binary linear sketch, i.e., a fixed number of parity measurements. We show that under a certain monotonicity condition, simply truncating the input by retaining a subset of its coordinates is optimal among all linear sketches of the same size. In particular, this applies to distributed testing of correlations of binary vectors when they are of opposite signs and evaluating unate Boolean functions on uniformly random inputs. Additionally, given that truncation is optimal, we immediately have lower bounds for arbitrary binary linear sketches. To illustrate this application, we consider the one-way communication complexity of a distributional graph connectivity problem on $n$ vertices and show that truncation, and hence all binary linear sketches, requires $Θ(n \log n)$ bits to achieve constant error, while a nonlinear protocol only requires $O(\log n)$ bits.