# Statistical inverse learning and ℓ1-regularization

arXiv: https://arxiv.org/abs/2607.07468 · HTML: https://mlsanity.com/paper/2607.07468/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 54 references that are in the map.
- First posted on arXiv 8 Jul 2026.

## Builds on (most cited first, 11 of 11)

- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Optimal rates for the regularized learning algorithms under general source condition](https://mlsanity.com/paper/1611.01900/) (2016)
- [Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems](https://mlsanity.com/paper/1902.05404/) (2019)
- [Convex regularization in statistical inverse learning problems](https://mlsanity.com/paper/2102.09526/) (2021)
- [Inverse learning in Hilbert scales](https://mlsanity.com/paper/2002.10208/) (2020)
- [Kernel regression, minimax rates and effective dimensionality: beyond the regular case](https://mlsanity.com/paper/1611.03979/) (2016)
- [Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform](https://mlsanity.com/paper/2302.03577/) (2023)
- [Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI](https://mlsanity.com/paper/2501.01929/) (2025)
- [Nonparametric goodness-of-fit testing for parametric covariate models in pharmacometric analyses](https://mlsanity.com/paper/2011.07539/) (2020)

## Abstract (arXiv)

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 ℓ1, and observations are generated through a possibly nonlinear forward operator A:ℓ1→ H, where H is a vector-valued reproducing kernel Hilbert space. We propose an ℓ1-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 ℓ1 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=Gcirc 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 kt 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.