Statistical inverse learning and ℓ1-regularization

Research Question and Approach

This paper addresses the challenge of recovering sparse signals from noisy, indirect, and finite observations in a statistical learning framework. While classical inverse problems often focus on deterministic noise, this work treats the problem as a statistical learning task where both the input sampling and the observational noise are random. The authors propose an L1-regularized empirical risk minimizer to promote sparsity in the recovered solution, extending the framework of vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHS) to handle nonlinear forward operators.

The authors provide a rigorous theoretical analysis of the proposed estimator, establishing almost-sure consistency as the sample size increases. They derive non-asymptotic, high-probability convergence rates in both prediction and L1-reconstruction norms. These rates are governed by two key parameters: the source smoothness index (r), which captures the regularity of the true solution via a variational source condition, and the effective dimension exponent (b), which characterizes the spectral decay of the covariance operator. By proving matching minimax lower bounds, the authors demonstrate that their derived convergence rates are optimal for this class of problems.

This work bridges the gap between deterministic regularization theory and modern statistical learning. By unifying concepts from compressed sensing, inverse problems, and kernel-based learning, the authors provide a framework that applies to complex, real-world scenarios such as reaction coefficient identification in elliptic PDEs and sparse computed tomography. The results offer a principled way to understand how sparsity-promoting penalties behave under random sampling, providing practitioners with concrete convergence guarantees for high-dimensional, ill-posed inverse problems.