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Tight bounds for minimum ℓ1-norm interpolation of noisy data
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:10572-10602, 2022.
Abstract
We provide matching upper and lower bounds of order σ2/log(d/n) for the prediction error of the minimum ℓ1-norm interpolator, a.k.a. basis pursuit. Our result is tight up to negligible terms when d≫n, and is the first to imply asymptotic consistency of noisy minimum-norm interpolation for isotropic features and sparse ground truths. Our work complements the literature on "benign overfitting" for minimum ℓ2-norm interpolation, where asymptotic consistency can be achieved only when the features are effectively low-dimensional.