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arXiv · 2305.11055

Small noise analysis for Tikhonov and RKHS regularizations

Quanjun Lang·Fei Lu
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Abstract

Regularization plays a pivotal role in ill-posed machine learning and inverse problems. However, the fundamental comparative analysis of various regularization norms remains open. We establish a small noise analysis framework to assess the effects of norms in Tikhonov and RKHS regularizations, in the context of ill-posed linear inverse problems with Gaussian noise. This framework studies the convergence rates of regularized estimators in the small noise limit and reveals the potential instability of the conventional L2-regularizer. We solve such instability by proposing an innovative class of adaptive fractional RKHS regularizers, which covers the L2 Tikhonov and RKHS regularizations by adjusting the fractional smoothness parameter. A surprising insight is that over-smoothing via these fractional RKHSs consistently yields optimal convergence rates, but the optimal hyper-parameter may decay too fast to be selected in practice.

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Quanjun Lang, Fei Lu. 2023-05-18. Small noise analysis for Tikhonov and RKHS regularizations. https://arxiv.org/abs/2305.11055

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