Optimal Rates For Regularization Of Statistical Inverse Learning Problems

TL;DR

This paper establishes minimax optimal convergence rates for spectral regularization methods in statistical inverse learning problems, accounting for unknown design distributions and noise, thereby advancing theoretical understanding of such inverse problems.

Contribution

It provides the first comprehensive analysis of optimal rates for both direct and inverse problems under general conditions, including unknown design distributions and source conditions.

Findings

Achieved minimax optimal rates for a broad class of spectral regularization methods.

Explicitly characterized the dependency of convergence rates on noise variance and source set radius.

Extended previous results to more general and realistic inverse learning scenarios.

Abstract

We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of $Af$) and the inverse (estimation of $f$) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations $n$ grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in $n$ but also in the explicit dependency of the constant factor in the variance of the noise and the radius of the source condition set.