Linear Sketching Over F-2

We initiate a systematic study of linear sketching over F-2. For a given Boolean function treated as f : F-2(n) -> F-2 a randomized F-2-sketch is a distribution M over d x n matrices with elements over F-2 such thatMx suffices for computing f(x) with high probability. Such sketches for d << n can be used to design small-space distributed and streaming algorithms.Motivated by these applications we study a connection between F-2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F-2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F-2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F-2-sketching is optimal (up to constant factors) for uniformly distributed inputs.Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F-2 can be constructed as F-2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length (O) over tilde (n) constructed through uniformly random updates.