  Let $f:\{-1,1\}^n$ be a polynomial with at most $s$ non-zero real
coefficients. We give an algorithm for exactly reconstructing f given random
examples from the uniform distribution on $\{-1,1\}^n$ that runs in time
polynomial in $n$ and $2s$ and succeeds if the function satisfies the unique
sign property: there is one output value which corresponds to a unique set of
values of the participating parities. This sufficient condition is satisfied
when every coefficient of f is perturbed by a small random noise, or satisfied
with high probability when s parity functions are chosen randomly or when all
the coefficients are positive. Learning sparse polynomials over the Boolean
domain in time polynomial in $n$ and $2s$ is considered notoriously hard in the
worst-case. Our result shows that the problem is tractable for almost all
sparse polynomials. Then, we show an application of this result to hypergraph
sketching which is the problem of learning a sparse (both in the number of
hyperedges and the size of the hyperedges) hypergraph from uniformly drawn
random cuts. We also provide experimental results on a real world dataset.
