A Threshold Regularization Method for Inverse Problems

TL;DR

This paper introduces a novel regularization method for inverse problems that extends spectral cut-off to non-monotonic filters, demonstrating near-optimal performance and efficiency in noisy operator scenarios.

Contribution

It proposes a new regularization approach that generalizes spectral cut-off to non-monotonic sequences, with theoretical guarantees and extensions to noisy operators.

Findings

Method is nearly optimal under mild assumptions.

Extends to inverse problems with noisy operators.

Provides oracle inequalities and efficiency results.

Abstract

A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. However, these so-called filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. In this paper, we point out that in several cases, non-monotonic sequences of filters are more efficient. We study a regularization method that naturally extends the spectral cut-off procedure to non-monotonic sequences and provide several oracle inequalities, showing the method to be nearly optimal under mild assumptions. Then, we extend the method to inverse problems with noisy operator and provide efficiency results in a newly introduced conditional framework.