arXiv subjects

Harshit Bajpai

Publications and source records attributed to Harshit Bajpai.

6 recordsLinked to original sources

arXiv2026

Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule

We introduce two novel randomized iterative regularization frameworks, termed \texttt{RIGKT} and \texttt{RIAT}, for solving large-scale linear ill-posed inverse problems governed by systems of equations. The proposed methods combine randomized iterated Tikhonov regularization with Krylov subspace projection techniques, utilizing Golub--Kahan bidiagonalization for general rectangular systems (\texttt{RIGKT}) and Arnoldi decomposition for square systems (\texttt{RIAT}). Unlike existing deterministic schemes that rely on fixed iteration counts, our framework incorporates randomized equation selection, an adaptive step-size strategy, and a global, discrepancy-based a posteriori early-stopping rule tailored specifically to the stochastic setting. We present a comprehensive regularization analysis establishing Bregman-distance monotonicity, finite termination, exact-data convergence, and pathwise stability under noise. Furthermore, we prove that the stopped iterates converge almost surely and in the mean-square sense to the true solution, establishing a rigorous regularization property. To the best of our knowledge, this is the first theoretical framework to simultaneously account for randomization, Krylov-subspace dimension reduction, and implementable early stopping. Numerical experiments involving two-dimensional X-ray computed tomography (CT) and image deblurring demonstrate that \texttt{RIGKT} and \texttt{RIAT} reliably reconstruct structural features across various noise regimes.

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arXiv2026

On the convergence of an adaptive denoiser driven iterative regularization with early stopping

Solving inverse problems requires appropriate regularization techniques to ensure well-posedness and stability. In recent years, denoiser-driven methods have emerged as effective regularization strategies, achieving state-of-the-art performance in various imaging applications. However, their stability and convergence within iterative regularization frameworks remain largely unexplored. In this work, we extend the framework of Regularization by Denoising (RED) by introducing a novel denoiser-driven iterative regularization scheme, referred to as \texttt{DDIR}, that incorporates a new regularization functional based on averaged denoisers. The proposed approach employs an adaptive step-size strategy together with an \emph{a posteriori} stopping rule to ensure stability while alleviating oscillatory behavior and semi-convergence effects induced by noise. As our main theoretical contribution, we prove that the resulting reconstruction method constitutes a stable and convergent regularization scheme in the classical sense. To the best of our knowledge, this provides the first rigorous justification of \texttt{DDIR} within the framework of regularization theory. Finally, we demonstrate the performance of the proposed method through numerical experiments on image deblurring and phase retrieval Computed Tomography (CT) using three denoisers, namely median, TNRD, and TV proximal. The results highlight the effectiveness of the method in terms of reconstruction accuracy and computational efficiency.

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arXiv2026

Graph Laplacian assisted regularization method under noise level free heuristic and statistical stopping rule

In this work, we address the solution of both linear and nonlinear ill-posed inverse problems by developing a novel graph-based regularization framework, where the regularization term is formulated through an iteratively updated graph Laplacian. The proposed approach operates without prior knowledge of the noise level and employs two distinct stopping criteria namely, the heuristic rule and the statistical discrepancy principle. To facilitate the latter, we utilize averaged measurements derived from multiple repeated observations. We provide a detailed convergence analysis of the method in statistical prospective, establishing its stability and regularization properties under both stopping strategies. The algorithm begins with the computation of an initial reconstruction using any suitable techniques like Tikhonov regularization (Tik), filtered back projection (FBP) or total variation (TV), which is used as the foundation for generating the initial graph Laplacian. The reconstruction is made better step by step using an iterative process, during which the graph Laplacian is dynamically re-calibrated to reflect how the solution's structure is changing. Finally, we present numerical experiments on X-ray Computed Tomography (CT) and phase retrieval CT, demonstrating the effectiveness and robustness of the proposed method and comparing its reconstruction performance under both stopping rules.

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arXiv2025

On the convergence of iterative regularization method assisted by the graph Laplacian with early stopping

We present a data-assisted iterative regularization method for solving ill-posed inverse problems. The proposed approach, termed \texttt{IRMGL+$\Psi$}, integrates classical iterative techniques with a data-driven regularization term realized through an iteratively updated graph Laplacian. Our method commences by computing a preliminary solution using any suitable reconstruction method, which then serves as the basis for constructing the initial graph Laplacian. The solution is subsequently refined through an iterative process, where the graph Laplacian is simultaneously recalibrated at each step to effectively capture the evolving structure of the solution. A key innovation of this work lies in the formulation of this iterative scheme and the rigorous justification of the classical discrepancy principle as a reliable early stopping criterion specifically tailored to the proposed method. Under standard assumptions, we establish stability and convergence results for the scheme when the discrepancy principle is applied. Furthermore, we demonstrate the robustness and effectiveness of our method through numerical experiments utilizing four distinct initial reconstructors $\Psi$: the adjoint operator (Adj), filtered back projection (FBP), total variation (TV) denoising, and standard Tikhonov regularization (Tik). It is observed that \texttt{IRMGL+Adj} demonstrates a distinct advantage over the other initializers, producing a robust and stable approximate solution directly from a basic initial reconstruction.

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arXiv2024

Hanke-Raus heuristic rule for iteratively regularized stochastic gradient descent

Over the past decade, stochastic algorithms have emerged as scalable and efficient tools for solving large-scale ill-posed inverse problems by randomly selecting subsets of equations at each iteration. However, due to the ill-posedness and measurement noise, these methods often suffer from oscillations and semi-convergence behavior, posing challenges in achieving stable and accurate reconstructions. This study proposes a novel variant of the stochastic gradient descent (SGD) approach, the iteratively regularized stochastic gradient descent (IRSGD) to address nonlinear ill-posed problems in Hilbert spaces. Under standard assumptions, we demonstrate that the mean square iteration error of the method tends to zero for exact data. In the presence of noisy data, we first propose a heuristic parameter choice rule (HPCR) and then apply the IRSGD method in combination with HPCR. Precisely, HPCR selects the regularization parameter without requiring any a-priori information of the noise level. We show that the method terminatesinfinitelymanystepsincaseofnoisydataandhasregularizingfeatures. Finally, some numerical experiments are performed to demonstrate the practical efficacy of the method.

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arXiv2024

Stochastic Data-Driven Bouligand Landweber Method for Solving Non-smooth Inverse Problems

In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an \textit{a-priori} stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by discussing an example that fulfills assumptions of the paper.

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