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regularization parameter estimation for feedforward neural networks

IEEE TRANSACTIONS ON SYSTEMS MAN AND CYBERNETICS PART B-CYBERNETICS 2003年第1期33卷 35-44页

作者： Guo, P Lyu, MR Chen, CLP Beijing Normal Univ Dept Comp Sci Beijing 100875 Peoples R China Chinese Univ Hong Kong Dept Comp Engn & Sci Shatin Hong Kong Peoples R China Univ Texas Dept Elect & Comp Engn San Antonio TX 78249 USA

Under the framework of the Kullback-Leibler (KL) distance, we show that a particular case of Gaussian probability function for feedforward neural networks (NNs) reduces into the first-order Tikhonov regularizer. The smooth parameter in kernel density estimation plays the role of the regularization parameter. Under some approximations, an estimation formula is derived for estimating regularization parameters based on training data sets. The similarity and difference of the obtained results are compared with other's work. Experimental results show that the estimation formula works well in the sparse and small training sample cases.

关键词： regularization parameter estimation small training data set Tikhonov regularizer

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regularization parameter estimation for Non-Negative Hyperspectral Image Deconvolution

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2016年第11期25卷 5316-5330页

作者： Song, Yingying Brie, David Djermoune, El-Hadi Henrot, Simon Univ Lorraine Ctr Rech Automat Nancy F-54506 Vandoeuvre Les Nancy France Univ Lorraine Ctr Natl Rech Sci F-54506 Vandoeuvre Les Nancy France Signal & Image Dept Grenoble Images Parole Signal Automat Lab F-38402 St Martin Dheres France

This paper aims at studying a method to automatically estimate the regularization parameters of non-negative hyperspectral image deconvolution methods. The deconvolution problem is formulated as a multi-objective optimization problem and the properties of the corresponding response surface are studied. Based on these properties, the minimum distance criterion (MDC) and the maximum curvature criterion (MCC) are proposed to estimate regularization parameters especially for the non-negativity constrained deconvolution problem. MDC has good theoretical properties (convexity and uniqueness) but requires to choose a reference point. On the contrary, MCC does not need to choose any reference point but does not have interesting theoretical properties. A grid-search-based approach to minimize the computational cost of MDC and MCC is proposed. It results in fast approaches to estimate the regularization parameters. Based on simulated 2D images, the proposed approaches are compared with the state-of-the-art methods, confirming the effectiveness of the MDC and MCC for the non-negativity constrained image deconvolution problem. In the case of non-negative hyperpsectral image deconvolution, the fast MDC yields better performances than the fast MCC. An application to real-world hyperspectral fluorescence microscopy images is also provided;it confirms the superiority of MDC.

关键词： Non-negative hyperspectral image deconvolution multi-objective optimization regularization parameter estimation

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Optimal regularization parameter estimation for Spectral Regression Discriminant Analysis

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2009年第12期19卷 1921-1926页

作者： Chen, Wei Shan, Caifeng de Haan, Gerard Philips Res Labs NL-5656 AE Eindhoven Netherlands

Spectral regression discriminant analysis (SRDA) is an efficient subspace learning method proposed recently. One important unsolved issue of SRDA is how to automatically determine an appropriate regularization parameter. In this letter, we present a method to estimate the optimal regularization parameter for SRDA. We test our method in different applications including head pose estimation, face recognition, and text categorization. Our extensive experiments evidently illustrate the effectiveness and efficiency of our approach.

关键词： regularization parameter estimation spectral regression discriminant analysis subspace learning

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Hyperspectral Image Super-Resolution via Deep Prior regularization With parameter estimation

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2022年第4期32卷 1708-1723页

作者： Wang, Xiuheng Chen, Jie Wei, Qi Richard, Cedric Northwestern Polytech Univ Sch Marine Sci & Technol Xian 710072 Peoples R China JP Morgan New York NY 10017 USA Univ Cote dAzur OCA CNRS F-06108 Nice France

Hyperspectral image (HSI) super-resolution is commonly used to overcome the hardware limitations of existing hyperspectral imaging systems on spatial resolution. It fuses a low-resolution (LR) HSI and a high-resolution (HR) conventional image of the same scene to obtain an HR HSI. In this work, we propose a method that integrates a physical model and deep prior information. Specifically, a novel, yet effective two-stream fusion network is designed to serve as a regularizer for the fusion problem. This fusion problem is formulated as an optimization problem whose solution can be obtained by solving a Sylvester equation. Furthermore, the regularization parameter is simultaneously estimated to automatically adjust contribution of the physical model and the learned prior to reconstruct the final HR HSI. Experimental results on both simulated and real data demonstrate the superiority of the proposed method over other state-of-the-art methods on both quantitative and qualitative comparisons.

关键词： Superresolution Hyperspectral imaging Optimization Tensors Spatial resolution Degradation Matrix decomposition HSI super-resolution deep learning fusion Sylvester equation regularization parameter estimation

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A New Method of regularization parameter estimation for Source Localization

A New Method of Regularization Parameter Estimation for Sour...

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2011 IEEE CIE International Conference on Radar(2011年IEEE国际雷达会议RADAR 2011)

作者： Juanru Huang Mei Dong Shili Li National Key Lab for Radar Signal Processing Xidian Univ. Xi’an 710071 China

ISBN: (纸本)9781424484447

The problem of estimating the regularization parameter for source localization in sparse-regularization framework is considered in this paper. We employ the distribution about every entry of the square of the Frobenius norm of noise to obtain a larger and more appropriate regularization parameter. The paper analyzes the reason that we can not simply set it equal to the square of the Frobenius norm of noise and presents the estimation in two practical cases: one works without taking singular value decomposition (SVD) of sensor outputs;the other works after that pretreatment for large data quantity. The simulation results demonstrate that the proposed method has many advantages, including enhancing resolution, effectively suppressing spurious peaks, improving robustness to noise, as well as increasing the number of resolvable sources.

关键词： sparse representation regularization parameter estimation source localization sensor array processing

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Efficient estimation of regularization parameters via downsampling and the singular value expansion

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BIT NUMERICAL MATHEMATICS 2017年第2期57卷 499-529页

作者： Renaut, Rosemary A. Horst, Michael Wang, Yang Cochran, Douglas Hansen, Jakob Arizona State Univ Sch Math & Stat Sci POB 871804 Tempe AZ 85287 USA Ohio State Univ Dept Math 100 Math Tower231 West 18th Ave Columbus OH 43210 USA Hong Kong Univ Sci & Technol Dept Math Kowloon Hong Kong Peoples R China Arizona State Univ Sch Elect Comp & Energy Engn Ira A Fulton Sch Engn POB 875706 Tempe AZ 85287 USA Univ Penn David Rittenhouse Lab 209 S 33rd St Philadelphia PA 19104 USA

The solution, , of the linear system of equations arising from the discretization of an ill-posed integral equation with a square integrable kernel H(s, t) is considered. The Tikhonov regularized solution approximating the Galerkin coefficients of f(t) is found as the minimizer of , where is given by the Galerkin coefficients of g(s). depends on the regularization parameter that trades off between the data fidelity and the smoothing norm determined by L, here assumed to be diagonal and invertible. The Galerkin method provides the relationship between the singular value expansion of the continuous kernel and the singular value decomposition of the discrete system matrix for square integrable kernels. We prove that the kernel maintains square integrability under left and right multiplication by bounded functions and thus the relationship also extends to appropriately weighted kernels. The resulting approximation of the integral equation permits examination of the properties of the regularized solution independent of the sample size of the data. We prove that consistently down sampling both the system matrix and the data provides a small scale system that preserves the dominant terms of the right singular subspace of the system and can then be used to estimate the regularization parameter for the original system. When g(s) is directly measured via its Galerkin coefficients the regularization parameter is preserved across resolutions. For measurements of g(s) a scaling argument is required to move across resolutions of the systems when the regularization parameter is found using a regularization parameter estimation technique that depends on the knowledge of the variance in the data. Numerical results illustrate the theory and demonstrate the practicality of the approach for regularization parameter estimation using generalized cross validation, unbiased predictive risk estimation and the discrepancy principle applied to both the system of equations, and to the regulari

关键词： Singular value expansion Singular value decomposition Ill-posed inverse problem Tikhonov regularization regularization parameter estimation

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How to Estimate the regularization parameter for Spectral Regression Discriminant Analysis and its Kernel Version?

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2014年第2期24卷 211-223页

作者： Gui, Jie Sun, Zhenan Cheng, Jun Ji, Shuiwang Wu, Xindong Chinese Acad Sci Hefei Inst Intelligent Machines Hefei 230031 Anhui Peoples R China Chinese Acad Sci Inst Automat Natl Lab Pattern Recognit Ctr Res Intelligent Percept & Comp Beijing 100190 Peoples R China Chinese Acad Sci Natl Lab Pattern Recognit Ctr Res Intelligent Percept & Comp Inst Automat Beijing 100190 Peoples R China Chinese Acad Sci Shenzhen Inst Adv Technol Guangdong Prov Key Lab Robot & Intelligent Syst Shenzhen 518055 Peoples R China Old Dominion Univ Dept Comp Sci Norfolk VA USA Univ Vermont Dept Comp Sci Burlington VT 05401 USA

Spectral regression discriminant analysis (SRDA) has recently been proposed as an efficient solution to large-scale subspace learning problems. There is a tunable regularization parameter in SRDA, which is critical to algorithm performance. However, how to automatically set this parameter has not been well solved until now. So this regularization parameter was only set to be a constant in SRDA, which is obviously suboptimal. This paper proposes to automatically estimate the optimal regularization parameter of SRDA based on the perturbation linear discriminant analysis (PLDA). In addition, two parameter estimation methods for the kernel version of SRDA are also developed. One is derived from the method of optimal regularization parameter estimation for SRDA. The other is to utilize the kernel version of PLDA. Experiments on a number of publicly available databases demonstrate the effectiveness of the proposed methods for face recognition, spoken letter recognition, handwritten digit recognition, and text categorization.

关键词： Kernel methods perturbation linear discriminant analysis regularization parameter estimation spectral regression discriminant analysis

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Numerical Methods for parameter estimation in Poisson Data Inversion

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2015年第3期52卷 397-413页

作者： Zanni, Luca Benfenati, Alessandro Bertero, Mario Ruggiero, Valeria Univ Modena & Reggio Emilia Dipartimento Sci Fis Informat & Matemat I-41125 Modena Italy Univ Ferrara Dipartimento Matemat & Informat I-44122 Ferrara Italy Univ Genoa Dipartimento Informat Bioingn Robot & Ingn Sistem I-16146 Genoa Italy

In a regularized approach to Poisson data inversion, the problem is reduced to the minimization of an objective function which consists of two terms: a data-fidelity function, related to a generalized Kullback-Leibler divergence, and a regularization function expressing a priori information on the unknown image. This second function is multiplied by a parameter , sometimes called regularization parameter, which must be suitably estimated for obtaining a sensible solution. In order to estimate this parameter, a discrepancy principle has been recently proposed, that implies the minimization of the objective function for several values of . Since this approach can be computationally expensive, it has also been proposed to replace it with a constrained minimization, the constraint being derived from the discrepancy principle. In this paper we intend to compare the two approaches from the computational point of view. In particular, we propose a secant-based method for solving the discrepancy equation arising in the first approach;when this root-finding algorithm can be combined with an efficient solver of the inner minimization problems, the first approach can be competitive and sometimes faster than the second one.

关键词： Poisson noise regularization parameter estimation Discrepancy principle Hypersurface regularization Seminorm regularization

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HYBRID AND ITERATIVELY REWEIGHTED regularization BY UNBIASED PREDICTIVE RISK AND WEIGHTED GCV FOR PROJECTED SYSTEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2017年第2期39卷 B221-B243页

作者： Renaut, Rosemary A. Vatankhah, Saeed Ardestani, Vahid E. Arizona State Univ Sch Math & Stat Sci Tempe AZ 85287 USA Univ Tehran Inst Geophys Tehran Iran

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization, parameter for the projected problem by unbiased predictive risk estimation, generalized cross validation, and discrepancy principle techniques is investigated. It is shown that the regularized parameter obtained by the unbiased predictive risk estimator can provide a good estimate which can be used for a full problem that is moderately to severely ill-posed. A similar analysis provides the weight parameter for the weighted generalized cross validation such that the approach is also useful in these cases, and also explains why the generalized cross validation without weighting is not always useful. All results are independent of whether systems are over or underdetermined. Numerical simulations for standard one-dimensional test problems and two-dimensional data, for both image restoration and tomographic image reconstruction, support the analysis and validate the techniques. The size of the projected problem is found using an extension of a noise revealing function for the projected problem [I. Hnetynkova, M. Plesinger, and Z. Strakos, BIT Numer. Math., 49 (2009), pp. 669-696]. Furthermore, an iteratively reweighted regularization approach for edge preserving regularization is extended for projected systems, providing stabilization of the solutions of the projected systems and reducing dependence on the determination of the size of the projected subspace.

关键词： large-scale inverse problems Golub Kahan bidiagonalization regularization parameter estimation unbiased predictive risk estimator discrepancy principle generalized cross validation iteratively reweighted schemes

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Equivalent dipole source imaging of brain electric activity by means of parametric projection filter

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ANNALS OF BIOMEDICAL ENGINEERING 2001年第5期29卷 436-445页

作者： Hori, J He, B Univ Illinois Dept EECS Chicago IL 60607 USA Univ Illinois Dept Bioengn Chicago IL 60607 USA

In the present study, spatial filters for inverse estimation of an equivalent dipole layer from the scalp-recorded potentials have been explored for their suitability in achieving high-resolution electroencephalogram (EEG) imaging. The performance of the parametric projection filter (PPF), which we propose to use for high-resolution EEG imaging, has been evaluated by computer simulations in the presence of a priori information on noise. An inhomogeneous three-concentric-sphere head model was used in the present simulation study to represent the head volume conductor. An equivalent dipole layer was used to model brain electric sources and estimated From the scalp potentials. Various noise conditions were simulated and the parametric projection filter was compared with standard regularization procedures such as the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TKNV). The present simulation results suggest that the proposed method performs better than that of commonly used inverse regularization techniques, such as the general inverse using the TSVD and the TKNV, when the correlation between the original source distribution and the noise distribution is low, and performs similarly when the correlation is high. A method for determining the optimum regularization parameter, which can be applied to parametric inverse techniques, has also been developed. (C) 2001 Biomedical Engineering Society.

关键词： high-resolution EEG equivalent dipole source inverse problem parametric projection filter nonuniform noise noise covariance regularization parameter estimation

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