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1. Generalized cross validation for ℓp-ℓq minimization

   https://doi.org/10.1007/s11075-021-01087-9  

   Buccini, Alessandro; Reichel, Lothar (March 2021, Numerical Algorithms)

   null (Ed.)

   Abstract Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p -norm, and a regularization term, that is defined in terms of a q -norm. We allow 0 < p , q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed. 

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2. Regularized Momentum Iterative Hessian Sketch for Large Scale Linear System of Equations

   Ozaslan, Ibrahim Kurban; Pilanci, Mert; Arikan, Orhan (January 2020, International Conferene on Acoustics, Speech and Signal Processing)

   null (Ed.)

   In this article, Momentum Iterative Hessian Sketch (M-IHS) techniques, a group of solvers for large scale linear Least Squares (LS) problems, are proposed and analyzed in detail. The proposed techniques are obtained by incorporating the Heavy Ball Acceleration into the Iterative Hessian Sketch algorithm and they provide significant improvements over the randomized preconditioning techniques. Through the error analyses of the M-IHS variants, lower bounds on the sketch size for various randomized distributions to converge at a pre-determined rate with a constant probability are established. The bounds present the best results in the current literature for obtaining a solution approximation and they suggest that the sketch size can be chosen proportional to the statistical dimension of the regularized problem regardless of the size of the coefficient matrix. The statistical dimension is always smaller than the rank and it gets smaller as the regularization parameter increases. By using approximate solvers along with the iterations, the M-IHS variants are capable of avoiding all matrix decompositions and inversions, which is one of the main advantages over the alternative solvers such as the Blendenpik and the LSRN. Similar to the Chebyshev Semi-iterations, the M-IHS variants do not use any inner products and eliminate the corresponding synchronizations steps in hierarchical or distributed memory systems, yet the M-IHS converges faster than the Chebyshev Semi-iteration based solvers 

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3. Learning to Sense for Coded Diffraction Imaging

   https://doi.org/10.3390/s22249964  

   Hyder, Rakib; Cai, Zikui; Asif, M. Salman (December 2022, Sensors)

   In this paper, we present a framework to learn illumination patterns to improve the quality of signal recovery for coded diffraction imaging. We use an alternating minimization-based phase retrieval method with a fixed number of iterations as the iterative method. We represent the iterative phase retrieval method as an unrolled network with a fixed number of layers where each layer of the network corresponds to a single step of iteration, and we minimize the recovery error by optimizing over the illumination patterns. Since the number of iterations/layers is fixed, the recovery has a fixed computational cost. Extensive experimental results on a variety of datasets demonstrate that our proposed method significantly improves the quality of image reconstruction at a fixed computational cost with illumination patterns learned only using a small number of training images. 

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4. Momentum-Net: Fast and convergent iterative neural network for inverse problems

   https://doi.org/10.1109/TPAMI.2020.3012955  

   Chun, Il Yong; Huang, Zhengyu; Lim, Hongki; Fessler, Jeff (January 2020, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   Iterative neural networks (INN) are rapidly gaining attention for solving inverse problems in imaging, image processing, and computer vision. INNs combine regression NNs and an iterative model-based image reconstruction (MBIR) algorithm, often leading to both good generalization capability and outperforming reconstruction quality over existing MBIR optimization models. This paper proposes the first fast and convergent INN architecture, Momentum-Net, by generalizing a block-wise MBIR algorithm that uses momentum and majorizers with regression NNs. For fast MBIR, Momentum-Net uses momentum terms in extrapolation modules, and noniterative MBIR modules at each iteration by using majorizers, where each iteration of Momentum-Net consists of three core modules: image refining, extrapolation, and MBIR. Momentum-Net guarantees convergence to a fixed-point for general differentiable (non)convex MBIR functions (or data-fit terms) and convex feasible sets, under two asymptomatic conditions. To consider data-fit variations across training and testing samples, we also propose a regularization parameter selection scheme based on the “spectral spread” of majorization matrices. Numerical experiments for light-field photography using a focal stack and sparse-view computational tomography demonstrate that, given identical regression NN architectures, Momentum-Net significantly improves MBIR speed and accuracy over several existing INNs; it significantly improves reconstruction quality compared to a state-of-the-art MBIR method in each application 

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5. Dynamically accelerating the power iteration with momentum

   https://doi.org/10.1002/nla.2584  

   Austin, Christian; Pollock, Sara; Zhu, Yunrong (August 2024, Numerical Linear Algebra with Applications)

   In this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations. 

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