Statistical inverse learning and $\ell^1$-regularization

2026-07-08

AI Summary

This study addresses the statistical inverse problem of recovering sparse functions from finite, noisy, and indirect observations. The unknown function is modeled as an element of an $\ell^1$ space, with data generated via a nonlinear forward operator, and reconstruction is performed through $\ell^1$-regularized empirical risk minimization. Within the framework of reproducing kernel Hilbert spaces, the authors establish almost sure consistency and non-asymptotic high-probability convergence rates by combining approximation space theory with variational source conditions. Both the prediction error and the $\ell^1$ reconstruction error attain minimax optimality, and explicit convergence rates are derived for the filtered Radon transform. The theoretical results are successfully applied to elliptic PDE coefficient identification and sparse computed tomography imaging.

Abstract

We study the recovery of sparse functions from finite, noisy, and indirect observations in the framework of statistical inverse learning. The unknown is modeled as an element of $\ell^1$, and observations are generated through a possibly nonlinear forward operator $A:\ell^1\to H$, where $H$ is a vector-valued reproducing kernel Hilbert space. We propose an $\ell^1$-regularized empirical risk minimizer and develop a theoretical analysis of its statistical properties. Under mild assumptions, we establish almost-sure consistency and derive non-asymptotic high-probability convergence rates in both the prediction and $\ell^1$ reconstruction norms. The rates depend on the source smoothness parameter $r$, characterized by a variational source condition, and the effective dimension exponent $b$, describing the polynomial spectral decay of the covariance operator. We further prove matching minimax lower bounds, showing that the obtained convergence rates are optimal. To relate the theory to practical sparsity models, we consider finitely smoothing operators of the form $A=G\circ S$, where $S$ is a synthesis operator, and show that approximation-space assumptions imply the required variational source conditions. In particular, we prove that membership in the approximation space $k_t$ is equivalent to polynomial decay of the best $n$-term approximation error. Finally, we verify the assumptions for two representative inverse problems: reaction coefficient identification in elliptic PDEs and sparse computed tomography. For filtered Radon transforms, we derive explicit effective-dimension asymptotics, yielding concrete convergence rates for standard image models and sparsifying systems.

Problem

• statistical inverse learning
• sparse recovery
• ℓ¹-regularization
• nonlinear inverse problems
• noisy observations

Innovation

• statistical inverse learning
• ℓ¹-regularization
• sparsity recovery
• minimax optimality
• variational source condition