@inproceedings{Xiang2018,
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 = {Huang, Xiang and He, Kuan and Yoo, Seunghwan and Cossairt, Oliver and Katsaggelos, Aggelos and Ferrier, Nicola and Hereld, Mark},
booktitle = {2018 25th IEEE International Conference on Image Processing (ICIP)},
doi = {10.1109/ICIP.2018.8451710},
isbn = {978-1-4799-7061-2},
issn = {15224880},
keywords = {3d volumetric image reconstruction,Compressive sensing,Nonnegative sparse,Norm regularized optimization,Primal-dual preconditioned interior point method},
month = {oct},
pages = {1193--1197},
publisher = {IEEE},
title = {{An Interior Point Method for Nonnegative Sparse Signal Reconstruction}},
url = {https://ieeexplore.ieee.org/document/8451710/},
year = {2018}
}

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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. 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