arXiv subjects

Darrel K Joseph

Publications and source records attributed to Darrel K Joseph.

2 recordsLinked to original sources

arXiv2026

Regularization of Statistical Inverse Problems on Non-Reflexive Banach Spaces

Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $ρ$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.

math.NA↗

arXiv2026

Convergence Analysis of Statistical Inverse Problems on Reproducing Kernel Banach Spaces

Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $ρ$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.

math.NA↗