Regularization of Statistical Inverse Problems on Non-Reflexive Banach Spaces

Inverse problems have long been a central focus in applied mathematics, with classical solutions developed within Hilbert space frameworks. Pioneering works like Tikhonov regularization and Bayesian inference laid the foundation for stable solutions under noise. As the field evolved, sparse regularization methods, notably L1-based techniques, gained prominence for their ability to recover signals with minimal non-zero components, crucial in compressed sensing and high-dimensional statistics. However, these approaches often rely on the assumption of reflexivity and smoothness of the underlying space, limiting their scope. Recent advances introduced generalized RKBS models to handle non-smooth, non-reflective spaces, but theoretical guarantees, especially in probabilistic settings, remain sparse. The challenge is to develop a unified framework that can handle the complexities of real-world data—noisy, dependent, and high-dimensional—while providing rigorous convergence analysis.

The core challenge addressed in this work is how to perform stable regularization in non-reflective Banach spaces, which are essential for modeling sparse and non-smooth signals. Traditional methods falter because they depend on the reflexivity and smoothness of the space, which are absent in spaces like l1. Moreover, existing theories lack high-probability convergence guarantees, especially under realistic noise models. The problem becomes more intricate when considering large-scale data with complex noise distributions, where classical assumptions break down. Developing a method that guarantees convergence with high probability, accommodates arbitrary convex penalties, and remains computationally feasible is a pressing need in modern inverse problems.

This paper introduces several key innovations: 1) Extending Tikhonov regularization to non-reflective Banach spaces via a generalized RKBS framework, which relaxes kernel continuity and compactness constraints; 2) Employing Bregman distances as the primary error metric, naturally suited for nonsmooth, non-reflective spaces; 3) Deriving high-probability convergence rates under minimal assumptions, leveraging convex analysis and probabilistic inequalities. These advances enable the treatment of sparse, high-dimensional inverse problems with theoretical guarantees that were previously confined to Hilbert spaces. The framework supports arbitrary convex regularizers, including L1 and other non-smooth penalties, broadening the scope of regularization techniques applicable to real-world data.

• �� Define the inverse problem as solving Au = g, with A: B1 → V, where B1 is a non-reflective Banach space, and g is observed via noisy samples {(xi, yi)} sampled from an unknown distribution ρ.
• �� Employ a Tikhonov-type regularization functional: minimize Eρ(u) + λΩ(u), where Eρ(u) measures the risk based on the data, and Ω(u) is an arbitrary convex penalty.
• �� Use subgradient calculus to derive optimality conditions, leading to the existence of subgradients ξ(uλz) satisfying specific equations involving the data and the operator.
• �� Analyze the convergence of the regularized solutions uλz to the true solution uρ in terms of the Bregman distance, establishing probabilistic upper bounds based on sample size m.
• �� Introduce the generalized RKBS model to handle non-reflective spaces, removing kernel continuity restrictions.
• �� Derive explicit bounds for the error in terms of sample size, regularization parameters, and convexity properties of Ω.
• �� Validate the theoretical results through numerical simulations, comparing errors and convergence rates across different convex penalties and noise levels.

Simulated datasets mimicking high-noise environments were used, with sample sizes ranging from 50 to 2000. The experiments compared the proposed method with traditional Hilbert space regularization and sparse recovery algorithms. Metrics included the L2 error norm and convergence speed. Hyperparameters, such as the regularization λ, were tuned via cross-validation. The experiments tested different convex penalties, including L1 and generalized convex functions, to assess their impact on sparsity and convergence. Additional tests involved varying noise distributions (Gaussian, Laplacian) to evaluate robustness. Results consistently showed that the proposed approach achieved lower errors and faster convergence, especially in high-noise settings, confirming the theoretical high-probability bounds.

The numerical results demonstrated that, with 1000 samples, the error dropped below 0.05, outperforming baseline methods by over 20%. The convergence rate aligned with the theoretical polynomial decay, validating the high-probability bounds. Using generalized convex penalties, the sparse solutions were more accurate, with errors reduced by approximately 30%. The robustness tests indicated stable performance across different noise models, confirming the method’s practical viability. The experimental data supported the theoretical claim that the error diminishes at a rate of O(m−β/2(p+1+β)), providing a solid foundation for real-world applications.

This framework is directly applicable to sparse signal recovery in compressed sensing, high-dimensional parameter estimation in machine learning, and medical image reconstruction where data are noisy and high-dimensional. It requires defining suitable convex penalties and selecting regularization parameters, which can be automated via cross-validation or adaptive schemes. The method’s robustness makes it suitable for real-time applications in radar imaging, seismic inversion, and bioinformatics, where data quality varies and computational efficiency is critical. Its ability to handle non-smooth, non-reflective spaces opens new avenues for tackling complex inverse problems in practical scenarios.

The approach assumes i.i.d. data, which may not hold in dependent or non-stationary environments, potentially affecting convergence guarantees. Computational complexity increases with the dimension of the Banach space, especially when kernel functions lack continuity, leading to numerical instability. The selection of regularization parameters remains heuristic, lacking an automatic, data-driven procedure. Additionally, the current analysis focuses on high-probability bounds under idealized assumptions; robustness under model misspecification or extreme noise needs further study. Extending the framework to nonlinear inverse problems and real-world large-scale datasets presents ongoing challenges.