Small noise analysis for Tikhonov and RKHS regularizations

• URL: http://arxiv.org/abs/2305.11055v2
• Date: Wed, 4 Sep 2024 02:24:25 GMT
• Title: Small noise analysis for Tikhonov and RKHS regularizations
• Authors: Quanjun Lang, Fei Lu,
• Abstract summary: We establish a small noise analysis framework to assess the effects of norms in Tikhonov and RKHS regularizations. This framework studies the convergence rates of regularized estimators in the small noise limit and reveals the potential instability of the conventional L2-regularizer. A surprising insight is that over-smoothing via these fractional RKHSs consistently yields optimal convergence rates.
• Score: 0.8133739801185272
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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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