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The Tikhonov regularization theory converts ill-posed inverse problems into well-posed problems by putting penalty on the solution sought. Instead of solving an inverse problem, the regularization theory minimizes a weighted sum of "data error" and "penalty" function, and it has been successfully applied to a variety of problems including tomography, inverse scattering, detection of radiation sources and early vision algorithms. Since the function to be minimized is a weighted sum of functions, one should estimate appropriate weights. This is a problem of hyperparameter estimation and a vast literature exists. Another problem is how one should compare a particular penalty function (regularizer) with another. This is a special class of model comparison problems which are generally difficult. A Hierarchical Bayesian scheme is proposed with multiple hyperparameters in order to cope with data containing subsets which consist of different degree of smoothness. The scheme outperforms the previous scheme with single hyperparameter.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E83-A No.8 pp.1641-1650

- Publication Date
- 2000/08/25

- Publicized

- Online ISSN

- DOI

- Type of Manuscript
- Special Section PAPER (Special Section on Digital Signal Processing)

- Category
- Applications of Signal Processing

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Ryo TAKEUCHI, Susumu NAKAZAWA, Kazuma KOIZUMI, Takashi MATSUMOTO, "A Hierarchical Bayesian Approach to Regularization Problems with Multiple Hyperparameters" in IEICE TRANSACTIONS on Fundamentals,
vol. E83-A, no. 8, pp. 1641-1650, August 2000, doi: .

Abstract: The Tikhonov regularization theory converts ill-posed inverse problems into well-posed problems by putting penalty on the solution sought. Instead of solving an inverse problem, the regularization theory minimizes a weighted sum of "data error" and "penalty" function, and it has been successfully applied to a variety of problems including tomography, inverse scattering, detection of radiation sources and early vision algorithms. Since the function to be minimized is a weighted sum of functions, one should estimate appropriate weights. This is a problem of hyperparameter estimation and a vast literature exists. Another problem is how one should compare a particular penalty function (regularizer) with another. This is a special class of model comparison problems which are generally difficult. A Hierarchical Bayesian scheme is proposed with multiple hyperparameters in order to cope with data containing subsets which consist of different degree of smoothness. The scheme outperforms the previous scheme with single hyperparameter.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e83-a_8_1641/_p

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@ARTICLE{e83-a_8_1641,

author={Ryo TAKEUCHI, Susumu NAKAZAWA, Kazuma KOIZUMI, Takashi MATSUMOTO, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={A Hierarchical Bayesian Approach to Regularization Problems with Multiple Hyperparameters},

year={2000},

volume={E83-A},

number={8},

pages={1641-1650},

abstract={The Tikhonov regularization theory converts ill-posed inverse problems into well-posed problems by putting penalty on the solution sought. Instead of solving an inverse problem, the regularization theory minimizes a weighted sum of "data error" and "penalty" function, and it has been successfully applied to a variety of problems including tomography, inverse scattering, detection of radiation sources and early vision algorithms. Since the function to be minimized is a weighted sum of functions, one should estimate appropriate weights. This is a problem of hyperparameter estimation and a vast literature exists. Another problem is how one should compare a particular penalty function (regularizer) with another. This is a special class of model comparison problems which are generally difficult. A Hierarchical Bayesian scheme is proposed with multiple hyperparameters in order to cope with data containing subsets which consist of different degree of smoothness. The scheme outperforms the previous scheme with single hyperparameter.},

keywords={},

doi={},

ISSN={},

month={August},}

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TY - JOUR

TI - A Hierarchical Bayesian Approach to Regularization Problems with Multiple Hyperparameters

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 1641

EP - 1650

AU - Ryo TAKEUCHI

AU - Susumu NAKAZAWA

AU - Kazuma KOIZUMI

AU - Takashi MATSUMOTO

PY - 2000

DO -

JO - IEICE TRANSACTIONS on Fundamentals

SN -

VL - E83-A

IS - 8

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - August 2000

AB - The Tikhonov regularization theory converts ill-posed inverse problems into well-posed problems by putting penalty on the solution sought. Instead of solving an inverse problem, the regularization theory minimizes a weighted sum of "data error" and "penalty" function, and it has been successfully applied to a variety of problems including tomography, inverse scattering, detection of radiation sources and early vision algorithms. Since the function to be minimized is a weighted sum of functions, one should estimate appropriate weights. This is a problem of hyperparameter estimation and a vast literature exists. Another problem is how one should compare a particular penalty function (regularizer) with another. This is a special class of model comparison problems which are generally difficult. A Hierarchical Bayesian scheme is proposed with multiple hyperparameters in order to cope with data containing subsets which consist of different degree of smoothness. The scheme outperforms the previous scheme with single hyperparameter.

ER -