Linear Sketching over 𝔽2

We initiate a systematic study of linear sketching over F2. For a given Boolean function f : {0, 1}n → {0, 1} a randomized F2-sketch is a distributionM over d×nmatrices with elements over F2 such that Mx suffices for computing f(x) with high probability. We study a connection between F2-sketching and a two-player one-way communication game for the corresponding XOR-function. Our results show that this communication game characterizes F2-sketching under the uniform distribution (up to dependence on error). Implications of this result include: 1) a composition theorem for F2-sketching complexity of a recursive majority function, 2) a tight relationship between F2-sketching complexity and Fourier sparsity, 3) lower bounds for a certain subclass of symmetric functions. We also fully resolve a conjecture of Montanaro and Osborne regarding one-way communication complexity of linear threshold functions by designing an F2sketch of optimal size. Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F2 can be constructed as F2-sketches for the uniform distribution with only a minor loss. 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 doesn’t require the stream length to be triply exponential in n and holds for streams of length Õ(n) constructed through uniformly random updates. Finally, we state a conjecture that asks whether optimal one-way communication protocols for XOR-functions can be constructed as F2-sketches with only a small loss. ∗University of Pennsylvania, [email protected] †Massachusetts Institute of Technology, [email protected]. E.M. acknowledges the support of grant N00014-16-12227 from Office of Naval Research and of NSF award CCF 1320105 as well as support from Simons Think Tank on Geometry & Algorithms. ‡Indiana University, Bloomington [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 4 of Report No. 174 (2016)