Randomized least-squares with minimal oversampling and interpolation in general spaces
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In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected L^2 sense, as soon as the sample size m is larger than the dimension n of the approximation space by a constant ratio. On the other hand, for m=n, we obtain an interpolation strategy with a stability factor of order n. The proposed sampling algorithms are greedy procedures based on arXiv:0808.0163 and arXiv:1508.03261, with polynomial computational complexity.
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