Statistical inverse learning with ℓ¹-regularization achieves optimal sparse recovery

The paper tackles the recovery of sparse functions from finite, noisy, and indirect observations—a core challenge in inverse problems like medical imaging and PDE parameter identification. The authors model the unknown as an element of ℓ¹ and propose an ℓ¹-regularized empirical risk minimizer. Under mild assumptions, they prove almost-sure consistency and derive non-asymptotic high-probability convergence rates for both prediction error and ℓ¹ reconstruction error. These rates depend on two key parameters: the source smoothness r (characterized by a variational source condition) and the effective dimension exponent b (describing the polynomial spectral decay of the covariance operator). They further prove matching minimax lower bounds, confirming that the rates are optimal.

The theoretical framework is linked to practical sparsity models by considering finitely smoothing operators of the form A = G ∘ S, where S is a synthesis operator. The authors show that approximation-space assumptions imply the required variational source conditions, equivalent to polynomial decay of the best n-term approximation error. They verify the assumptions on two representative inverse problems: reaction coefficient identification in elliptic PDEs and sparse computed tomography. For filtered Radon transforms, they derive explicit effective-dimension asymptotics, yielding concrete convergence rates for standard image models and sparsifying systems. This work provides a rigorous foundation for ℓ¹-regularization in statistical inverse learning, bridging theory and real-world applications.