An unbiased m-sparsification of a vector$p$E Rnis a random vector$Q$∊ Rnwith mean$p$that has at most m$n$nonzero coordinates. Unbiased sparsification compresses the original vector without introducing bias; it arises in various contexts, such as in federated learning and sampling sparse probability distributions. Ideally, unbiased sparsification should also minimize the expected value of a divergence function Div(Q, p) that measures how far away$Q$is from the original$p$. If$Q$is optimal in this sense, then we call it efficient. Our main results describe efficient unbiased sparsifications for divergences that are either permutation-invariant or additively separable. Surprisingly, the characterization for permutation-invariant divergences is robust to the choice of divergence function, in the sense that our class of optimal$Q$for squared Euclidean distance coincides with our class of optimal$Q$for Kullback-Leibler divergence, or indeed any of a wide variety of divergences.

In this paper, we prove that bent functions can be precisely characterized in terms of a special class of strongly regular graphs, thus providing a positive answer to a question raised in the paper by A. Bernasconi and B. Codenotti (1999).