Efficient Sampling for Learning Sparse Additive Models in High Dimensions

We consider the problem of learning sparse additive models, i.e., functions of the form: f(\vecx) = \sum_{l \in S} \phi_{l}(x_l), \vecx \in \matR^d from point queries of f. Here S is an unknown subset of coordinate variables with \abs{S} = k \ll d. Assuming \phi_l's to be smooth, we propose a set of points at which to sample f and an efficient randomized algorithm that recovers a \textit{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.