On the Equivalence between Quadrature Rules and Random Features
We show that kernel-based quadrature rules for computing integrals are a special case of random feature expansions for positive definite kernels for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation error, leading to both upper and lower bounds that are based solely on the eigenvalues of the associated integral operator and match up to logarithmic terms. In particular, we show that the upper bound may be obtained from independent and identically distributed samples from a known non-uniform distribution, while the lower bound if valid for any set of points. Applying our results to kernel-based quadrature, while our results are fairly general, we recover known upper and lower bounds for the special cases of Sobolev spaces. Moreover, our results extend to the more general problem of full function approximations (beyond simply computing an integral), with results in L2- and L∞-norm that match known results for special cases. Applying our results to random features, we show an improvement of the number of random features needed to preserve the generalization guarantees for learning with Lipschitz-continuous losses.
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