arXiv subjects

Abinash Nayak

Publications and source records attributed to Abinash Nayak.

7 recordsLinked to original sources

arXiv2025

A Gradient-thresholding Algorithm for Sparse Regularization

Inverse problems arise in a wide spectrum of applications in fields ranging from engineering to scientific computation. Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, such as Tikhonov-type regularization methods or iterative regularization methods, which are a necessity in most of the inverse problems. In the last few decades, regularization methods motivating sparsity has been the focus of research, due to the high dimensionalty of the real-life data, and $\mathcal{L}^1$-regularization methods (such as LASSO or FISTA) has been in its center (due to their computational simplicity). In this paper we propose a new (semi-) iterative regularization method which is not only simpler than the mentioned algorithms but also yields better results, in terms of accuracy and sparsity of the recovered solution. Furthermore, we also present a very effective and practical stopping criterion to choose an appropriate regularization parameter (here, it's iteration index) so as to recover a regularized (sparse) solution. To illustrate the computational efficiency of this algorithm we apply it to numerically solve the image deblurring problem and compare our results with certain standard regularization methods, like total variation, FISTA, LSQR etc.

math.NA↗

arXiv2022

Regularizing Instabilities in Image Reconstruction Arising from Learned Denoisers

It's well-known that inverse problems are ill-posed and to solve them meaningfully one has to employ regularization methods. Traditionally, popular regularization methods have been the penalized Variational approaches. In recent years, the classical regularized-reconstruction approaches have been outclassed by the (deep-learning-based) learned reconstruction algorithms. However, unlike the traditional regularization methods, the theoretical underpinnings, such as stability and regularization, have been insufficient for such learned reconstruction algorithms. Hence, the results obtained from such algorithms, though empirically outstanding, can't always be completely trusted, as they may contain certain instabilities or (hallucinated) features arising from the learned process. In fact, it has been shown that such learning algorithms are very susceptible to small (adversarial) noises in the data and can lead to severe instabilities in the recovered solution, which can be quite different than the inherent instabilities of the ill-posed (inverse) problem. Whereas, the classical regularization methods can handle such (adversarial) noises very well and can produce stable recovery. Here, we try to present certain regularization methods to stabilize such (unstable) learned reconstruction methods and recover a regularized solution, even in the presence of adversarial noises. For this, we need to extend the classical notion of regularization and incorporate it in the learned reconstruction algorithms. We also present some regularization techniques to regularize two of the most popular learning reconstruction algorithms, the Learned Post-Processing Reconstruction and the Learned Unrolling Reconstruction.

cs.LG↗

arXiv2021

Instabilities in Plug-and-Play (PnP) algorithms from a learned denoiser

It's well-known that inverse problems are ill-posed and to solve them meaningfully, one has to employ regularization methods. Traditionally, popular regularization methods are the penalized Variational approaches. In recent years, the classical regularization approaches have been outclassed by the so-called plug-and-play (PnP) algorithms, which copy the proximal gradient minimization processes, such as ADMM or FISTA, but with any general denoiser. However, unlike the traditional proximal gradient methods, the theoretical underpinnings, convergence, and stability results have been insufficient for these PnP-algorithms. Hence, the results obtained from these algorithms, though empirically outstanding, can't always be completely trusted, as they may contain certain instabilities or (hallucinated) features arising from the denoiser, especially when using a pre-trained learned denoiser. In fact, in this paper, we show that a PnP-algorithm can induce hallucinated features, when using a pre-trained deep-learning-based (DnCNN) denoiser. We show that such instabilities are quite different than the instabilities inherent to an ill-posed problem. We also present methods to subdue these instabilities and significantly improve the recoveries. We compare the advantages and disadvantages of a learned denoiser over a classical denoiser (here, BM3D), as well as, the effectiveness of the FISTA-PnP algorithm vs. the ADMM-PnP algorithm. In addition, we also provide an algorithm to combine these two denoisers, the learned and the classical, in a weighted fashion to produce even better results. We conclude with numerical results which validate the developed theories.

cs.CV↗

arXiv2021

Interpretation of Plug-and-Play (PnP) algorithms from a different angle

It's well-known that inverse problems are ill-posed and to solve them meaningfully one has to employ regularization methods. Traditionally, the most popular regularization approaches are Variational-type approaches, i.e., penalized/constrained functional minimization. In recent years, the classical regularization approaches have been replaced by the so-called plug-and-play (PnP) algorithms, which copies the proximal gradient minimization processes, such as ADMM or FISTA, but with any general denoiser. However, unlike the traditional proximal gradient methods, the theoretical analysis and convergence results have been insufficient for these PnP-algorithms. Hence, the results from these algorithms, though empirically outstanding, are not well-defined, in the sense of, being a minimizer of a Variational problem. In this paper, we address this question of "well-definedness", but from a different angle. We explain these algorithms from the viewpoint of a semi-iterative regularization method. In addition, we expand the family of regularized solutions, corresponding to the classical semi-iterative methods, to further generalize the explainability of these algorithms, as well as, enhance the recovery process. We conclude with several numerical results which validate the developed theories and reflect the improvements over the traditional PnP-algorithms, such as ADMM-PnP and FISTA-PnP.

math.OC↗

arXiv2021

Smoothing $\mathcal{L}^2$ gradients in iterative regularization

Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, which are a necessity due to the ill-posedness of inverse problems. Tikhonov-type regularization methods are very popular in this regard. However, its direct implementation for large-scale linear or non-linear problems is a non-trivial task. In such scenarios, iterative regularization methods usually serve as a better alternative. In this paper we propose a new iterative regularization method which uses descent directions, different from the usual gradient direction, that enable a more smoother and effective recovery than the later. This is achieved by transforming the original noisy gradient, via a smoothing operator, to a smoother gradient, which is more robust to the noise present in the data. It is also shown that this technique is very beneficial when dealing with data having large noise level. To illustrate the computational efficiency of this method we apply it to numerically solve some classical integral inverse problems, including image deblurring and tomography problems, and compare the results with certain standard regularization methods, such as Tikhonov, TV, CGLS, etc.

math.NA↗

arXiv2020

Parameter estimation in an elliptic problem

A new variational approach to solve the problem of estimating the (possibly discontinuous) coefficient functions $p$, $q$ and $f$ in elliptic equations of the form $-\nabla \cdot (p(x)\nabla u) + λq(x) u = f$, $x \in Ω\subset \mathbb{R}^n$, from a knowledge of the solutions $u_λ$.

math.NA↗

arXiv2020

A new regularization approach for numerical differentiation

The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to approximate the solution appropriately. The commonly practiced regularization methods are (external) parameter-based like Tikhonov regularization, which have certain inherent difficulties like choosing an optimal value of the regularization parameter, especially in the absence of noise information, which is a non-trivial task. An attractive alternative is iterative regularization, where one minimizes an associated functional in a regularized fashion (i.e., stopping at an appropriate iteration). However, in most of the regularization methods the associated minimizing functional contains the noisy data directly, and hence, the recovery gets affected. In this paper, we propose an iterative regularization method where the minimizing functional does not contain the noisy data directly. The advantage, in addition to circumventing the use of noisy data directly, is that the sequence of functions (or curves) constructed during the descent process does not converge strongly to the noise data, rather weakly and hence, avoids overfitting. Furthermore, this method is very robust to extreme noise level in the data as well as to errors with non-zero mean. To demonstrate the effectiveness of the new method we provide some examples comparing the numerical results obtained from our method with results obtained from some of the popular regularization methods such as Tikhonov regularization, total variation, smoothing spline and the polynomial regression method.

math.NA↗