  An unbiased $m$-sparsification of a vector $p\in \mathbb{R}^n$ is a random
vector $Q\in \mathbb{R}^n$ with 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
$\mathsf{Div}(Q,p)$ 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.
