Theory of Combinatorial Algorithms

We consider the problem of learning sparse additive models, i.e. functions of the form f(x)=\sum_{l\in S}\phi_l(x_l), x\in \R^d, from point queries of. Here, S is an unknown is an unknown ubset of coordinate variables with |S|=k\ll d. Assuming \phi_l's to be smooth, we propose a set of points at which to sample and an efficient randomized algorithm that recovers a uniform approximation to each unknown \phi_l. We provide a rigorous theoretical analysis of our scheme along with sample complexity bounds. Our algorithm utilizes recent results from compressive sensing theory along with a novel convex quadratic program for recovering robust uniform approximations to univariate functions, from point queries corrupted with arbitrary bounded noise. Lastly we theoretically analyze the impact of noise -- either arbitrary but bounded, or stochastic -- on the performance of our algorithm.