Algorithms for Learning Sparse Additive Models with Interactions in High Dimensions
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A function f: R^d →R is a Sparse Additive Model (SPAM), if it is of the form f(x) = ∑_l ∈Sϕ_l(x_l) where S⊂ [d], |S| ≪ d. Assuming ϕ's, S to be unknown, there exists extensive work for estimating f from its samples. In this work, we consider a generalized version of SPAMs, that also allows for the presence of a sparse number of second order interaction terms. For some S_1 ⊂ [d], S_2 ⊂[d] 2, with |S_1| ≪ d, |S_2| ≪ d^2, the function f is now assumed to be of the form: ∑_p ∈S_1ϕ_p (x_p) + ∑_(l,l^') ∈S_2ϕ_(l,l^') (x_l,x_l^'). Assuming we have the freedom to query f anywhere in its domain, we derive efficient algorithms that provably recover S_1,S_2 with finite sample bounds. Our analysis covers the noiseless setting where exact samples of f are obtained, and also extends to the noisy setting where the queries are corrupted with noise. For the noisy setting in particular, we consider two noise models namely: i.i.d Gaussian noise and arbitrary but bounded noise. Our main methods for identification of S_2 essentially rely on estimation of sparse Hessian matrices, for which we provide two novel compressed sensing based schemes. Once S_1, S_2 are known, we show how the individual components ϕ_p, ϕ_(l,l^') can be estimated via additional queries of f, with uniform error bounds. Lastly, we provide simulation results on synthetic data that validate our theoretical findings.
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