Adaptive minimax optimality in statistical inverse problems via SOLIT - Sharp Optimal Lepskii-Inspired Tuning
2023-04-20 | preprint. A publication with affiliation to the University of Göttingen.
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- Authors
- Li, Housen ; Werner, Frank
- Abstract
- We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form $\hat f_\alpha = q_\alpha \left(T^*T\right)T^*Y$, where $Y$ is the available data, $T$ the forward operator, $\left(q_\alpha\right)_{\alpha \in \mathcal A}$ an ordered filter, and $\alpha$ > 0 a regularization parameter. Whenever such a method is used in practice, $\alpha$ has to be chosen appropriately. Typically, the aim is to find or at least approximate the best possible $\alpha$ in the sense that mean squared error (MSE) $\mathbb E [\Vert \hat f_\alpha - f^\dagger\Vert^2]$ w.r.t.~the true solution $f^\dagger$ is minimized. In this paper, we introduce the Sharp Optimal Lepski\u{\i}-Inspired Tuning (SOLIT) method, which yields an a posteriori parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on $Y$ and the noise level $\sigma$ as well as the operator $T$ and the filter $\left(q_\alpha\right)_{\alpha \in \mathcal A}$ and does not require any problem-dependent tuning of further parameters. We prove an oracle inequality for the corresponding MSE in a general setting and derive the rates of convergence in different scenarios. By a careful analysis we show that no other a posteriori parameter choice rule can yield a better performance in terms of the convergence rate of the MSE. In particular, our results reveal that the typical understanding of Lepskiii-type methods in inverse problems leading to a loss of a log factor is wrong. In addition, the empirical performance of SOLIT is examined in simulations.
- Issue Date
- 20-April-2023
- Project
- SFB 1456: Mathematik des Experiments: Die Herausforderung indirekter Messungen in den Naturwissenschaften
SFB 1456 | Cluster B: Data with Incomplete Information
SFB 1456 | Cluster B | B04: Collective dynamics of ion channels: statistical modeling and analysis
EXC 2067: Multiscale Bioimaging - Organization
- Institut für Mathematische Stochastik
- Working Group
- RG Li
- Extent
- 26
- Language
- English