Hankel matrix nuclear norm regularized tensor completion for *N*-dimensional exponential signals (中文，English)

Jiaxi Ying^{1}, Hengfa Lu^{1}, Qingtao Wei^{4}, Jian-Feng Cai^{2}, Di Guo^{3}, Jihui Wu^{4}, Zhong Chen^{1}, Xiaobo Qu^{1,*}

1 Department of Electronic Science, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen, China

2 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China

3 School of Computer and Information Engineering, Xiamen University of Technology, Xiamen, China

4
School of Life Sciences, University of Science and Technology of China, Hefei, China.

* Email: quxiaobo <at> xmu.edu.cn or quxiaobo2009 <at> gmail.com.

Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology, and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired, and how to recover the full signal becomes an active research topic. However, existing exponential signal recovery, such as low rank Hankel matrix recovery [1] and enhanced matrix recovery, cannot efficiently recover *N*-dimensional exponential signals with *N* ≥ 3.

In this work, we study the problem of recovering *N*-dimensional (particularly *N* ≥ 3) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC tensor structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. A numerical algorithm is developed to solve the proposed model and its convergence is theoretically analyzed.

Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.

This work aims to reconstruct *N*-dimensional (particularly *N* ≥ 3) exponential signals. Figure 1, for example, shows a graphical illustration of a 3-dimensional exponential signal.

We proposed the following reconstruction model

where lambda is a regularization parameter that trades off the nuclear norm against the data consistency and is a linear operator, mapping an vector to a Hankel matrix.

To solve the reconstruction model above, we develop an algorithm based on alternating direction method of multipliers (ADMM). We demonstrate that the sequence generated by the algorithm converges. Furthermore, if we further impose some condition on the Lagrange multipliers, then the limit is a critical point.

We apply HMRTC to recover a 3-D simulated data and protein RNA spectrum. Figure 2 shows that HMRTC yields an average relative reconstruction error that is much smaller than ADM-TR and WCP. Figure 3 shows the ^{1}H-^{13}C and ^{1}H-^{15}N skyline projection spectra of HNCO spectrum, which indicates that HMRTC can achieve high quality of reconstruction obtained from only 10% of the fully sampled.

The demo code of HMRTC can be downloaded here.

Dataset: The 3D HNCO can be download here.

A 3-D HNCO spectrum is tested and its sample is the U-[^{15}N, ^{13}C] RNA recognition motifs domain of protein RNA binding motif protein 5, which is a component of the spliceosome A-complex. The ADM-TR, WCP and HMRTC are compared in recovering this 3-D spectrum with the size of 64 ×128×512 from a 3-D Poisson-gap nonuniformly sampled time domain data. All the spectra are processed in NMRPipe using a routine processing manner.

*Please cite this paper when using the HMRTC code. This citation is : *

Jiaxi Ying, Hengfa Lu, Qingtao Wei, Jian-Feng Cai, Di Guo, Jihui Wu, Zhong Chen, Xiaobo Qu*, Hankel matrix nuclear norm regularized tensor completion for *N*-dimensional exponential signals. * IEEE Transactions on Signal Processing*, 65(14): 3702-3717, 2017.

This work was partially supported by NSFC (61571380, 61672335, 61601276 and U1632274), Important Joint Research Project on Major Diseases of Xiamen City (3502Z20149032), Fundamental Research Funds for the Central Universities (20720150109) and the Natural Science Foundation of Fujian Province of China (2015J01346 and 2016J05205). The authors would like to thank Silvia Gandy, Gongguo Tang, Ji Liu, Xinhua Zhang and Andreas Jakobsson for sharing codes for comparisons and Weiyu Xu for helpful discussions. The authors also appreciate reviewers and editors for their constructive comments.

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