Learning Fourier Sparse Set Functions

Can we learn a sparse graph from observing the value of a few random cuts? This and more general problems can be reduced to the challenge of learning set functions known to have sparse Fourier support contained in some collection P. We prove that if we choose O(k log |P|) sets uniformly at random, then with high probability, observing any k-sparse function on those sets is sufficient to recover that function exactly. We further show that other properties, such as symmetry or submodularity imply structure in the Fourier spectrum, which can be exploited to further reduce sample complexity. One interesting special case is that it suffices to observe O(|E| log (|V |)) values of a cut function to recover a graph. We demonstrate the effectiveness of our results on two realworld reconstruction problems: graph sketching and obtaining fast approximate surrogates to expensive submodular objective functions.