An Interior Point Method for Nonnegative Sparse Signal Reconstruction

Xiang Huang, Kuan He, Seunghwan Yoo, Oliver Strides Cossairt, Aggelos K Katsaggelos, Nicola Ferrier, Mark Hereld

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We present a primal-dual interior point method (IPM) with a novel preconditioner to solve the ℓ 1 -norm regularized least square problem for nonnegative sparse signal reconstruction. IPM is a second-order method that uses both gradient and Hessian information to compute effective search directions and achieve super-linear convergence rates. It therefore requires many fewer iterations than first-order methods such as iterative shrinkage/thresholding algorithms (ISTA) that only achieve sub-linear convergence rates. However, each iteration of IPM is more expensive than in ISTA because it needs to evaluate an inverse of a Hessian matrix to compute the Newton direction. We propose to approximate each Hessian matrix by a diagonal matrix plus a rank-one matrix. This approximation matrix is easily invertible using the Sherman-Morrison formula, and is used as a novel preconditioner in a preconditioned conjugate gradient method to compute a truncated Newton direction. We demonstrate the efficiency of our algorithm in compressive 3D volumetric image reconstruction. Numerical experiments show favorable results of our method in comparison with previous interior point based and iterative shrinkage/thresholding based algorithms.
Original languageEnglish (US)
Title of host publication2018 25th IEEE International Conference on Image Processing (ICIP)
PublisherIEEE
ISBN (Electronic)978-1479970612
DOIs
StatePublished - 2018

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Signal reconstruction
Conjugate gradient method
Image reconstruction
Experiments

Cite this

Huang, X., He, K., Yoo, S., Cossairt, O. S., Katsaggelos, A. K., Ferrier, N., & Hereld, M. (2018). An Interior Point Method for Nonnegative Sparse Signal Reconstruction. In 2018 25th IEEE International Conference on Image Processing (ICIP) IEEE. https://doi.org/10.1109/ICIP.2018.8451710
Huang, Xiang ; He, Kuan ; Yoo, Seunghwan ; Cossairt, Oliver Strides ; Katsaggelos, Aggelos K ; Ferrier, Nicola ; Hereld, Mark. / An Interior Point Method for Nonnegative Sparse Signal Reconstruction. 2018 25th IEEE International Conference on Image Processing (ICIP). IEEE, 2018.
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title = "An Interior Point Method for Nonnegative Sparse Signal Reconstruction",
abstract = "We present a primal-dual interior point method (IPM) with a novel preconditioner to solve the ℓ 1 -norm regularized least square problem for nonnegative sparse signal reconstruction. IPM is a second-order method that uses both gradient and Hessian information to compute effective search directions and achieve super-linear convergence rates. It therefore requires many fewer iterations than first-order methods such as iterative shrinkage/thresholding algorithms (ISTA) that only achieve sub-linear convergence rates. However, each iteration of IPM is more expensive than in ISTA because it needs to evaluate an inverse of a Hessian matrix to compute the Newton direction. We propose to approximate each Hessian matrix by a diagonal matrix plus a rank-one matrix. This approximation matrix is easily invertible using the Sherman-Morrison formula, and is used as a novel preconditioner in a preconditioned conjugate gradient method to compute a truncated Newton direction. We demonstrate the efficiency of our algorithm in compressive 3D volumetric image reconstruction. Numerical experiments show favorable results of our method in comparison with previous interior point based and iterative shrinkage/thresholding based algorithms.",
author = "Xiang Huang and Kuan He and Seunghwan Yoo and Cossairt, {Oliver Strides} and Katsaggelos, {Aggelos K} and Nicola Ferrier and Mark Hereld",
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doi = "10.1109/ICIP.2018.8451710",
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Huang, X, He, K, Yoo, S, Cossairt, OS, Katsaggelos, AK, Ferrier, N & Hereld, M 2018, An Interior Point Method for Nonnegative Sparse Signal Reconstruction. in 2018 25th IEEE International Conference on Image Processing (ICIP). IEEE. https://doi.org/10.1109/ICIP.2018.8451710

An Interior Point Method for Nonnegative Sparse Signal Reconstruction. / Huang, Xiang; He, Kuan; Yoo, Seunghwan; Cossairt, Oliver Strides; Katsaggelos, Aggelos K; Ferrier, Nicola; Hereld, Mark.

2018 25th IEEE International Conference on Image Processing (ICIP). IEEE, 2018.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AU - Ferrier, Nicola

AU - Hereld, Mark

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N2 - We present a primal-dual interior point method (IPM) with a novel preconditioner to solve the ℓ 1 -norm regularized least square problem for nonnegative sparse signal reconstruction. IPM is a second-order method that uses both gradient and Hessian information to compute effective search directions and achieve super-linear convergence rates. It therefore requires many fewer iterations than first-order methods such as iterative shrinkage/thresholding algorithms (ISTA) that only achieve sub-linear convergence rates. However, each iteration of IPM is more expensive than in ISTA because it needs to evaluate an inverse of a Hessian matrix to compute the Newton direction. We propose to approximate each Hessian matrix by a diagonal matrix plus a rank-one matrix. This approximation matrix is easily invertible using the Sherman-Morrison formula, and is used as a novel preconditioner in a preconditioned conjugate gradient method to compute a truncated Newton direction. We demonstrate the efficiency of our algorithm in compressive 3D volumetric image reconstruction. Numerical experiments show favorable results of our method in comparison with previous interior point based and iterative shrinkage/thresholding based algorithms.

AB - We present a primal-dual interior point method (IPM) with a novel preconditioner to solve the ℓ 1 -norm regularized least square problem for nonnegative sparse signal reconstruction. IPM is a second-order method that uses both gradient and Hessian information to compute effective search directions and achieve super-linear convergence rates. It therefore requires many fewer iterations than first-order methods such as iterative shrinkage/thresholding algorithms (ISTA) that only achieve sub-linear convergence rates. However, each iteration of IPM is more expensive than in ISTA because it needs to evaluate an inverse of a Hessian matrix to compute the Newton direction. We propose to approximate each Hessian matrix by a diagonal matrix plus a rank-one matrix. This approximation matrix is easily invertible using the Sherman-Morrison formula, and is used as a novel preconditioner in a preconditioned conjugate gradient method to compute a truncated Newton direction. We demonstrate the efficiency of our algorithm in compressive 3D volumetric image reconstruction. Numerical experiments show favorable results of our method in comparison with previous interior point based and iterative shrinkage/thresholding based algorithms.

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Huang X, He K, Yoo S, Cossairt OS, Katsaggelos AK, Ferrier N et al. An Interior Point Method for Nonnegative Sparse Signal Reconstruction. In 2018 25th IEEE International Conference on Image Processing (ICIP). IEEE. 2018 https://doi.org/10.1109/ICIP.2018.8451710