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Maximum correntropy adaptation approach for robust compressive sensing reconstruction. (English) Zbl 07257011
Summary: Robust compressive sensing (CS) aims to recover the sparse signals from noisy measurements perturbed by non-Gaussian (i.e., heavy-tailed) noises, where traditional CS reconstruction algorithms may perform poorly owing to utilizing the $$l_2$$ error norm in optimization. In this paper, we propose a novel maximum correntropy adaptation approach for robust CS reconstruction. The task is formulated as a $$l_0$$ regularized maximum correntropy criterion $$(l_0$$-MCC) optimization problem and is solved by adaptive filtering approach. The proposed $$l_0$$-MCC algorithm has a simple algorithm structure and can adaptively estimate the sparsity. It can efficiently alleviate the negative impact of noise in the presence of large outliers. Moreover, a novel theoretical analysis on convergence of $$l_0$$-MCC is also performed. Furthermore, a mini-batch-based $$l_0$$-MCC (MB-$$l_0$$-MCC) algorithm is developed to speed up the convergence. Comparison with existing robust CS reconstruction algorithms is conducted via simulations, showing that the proposed methods can achieve better performance than existing state-of-the-art algorithms.

##### MSC:
 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 94A20 Sampling theory in information and communication theory 68T05 Learning and adaptive systems in artificial intelligence
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##### References:
 [1] Blumensath, T.; Davies, M. E., Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal., 27, 3, 265-274 (2008) · Zbl 1174.94008 [2] Candes, E. J.; Romberg, J.; Tao, T., Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52, 2, 489-509 (2006) · Zbl 1231.94017 [3] Candès, E. J., Compressive sampling, Proceedings of the International Congress of Mathematicians, 3, 1433-1452 (2006) · Zbl 1130.94013 [4] Candes, E. J.; Tao, T., Near-optimal signal recovery from random projections: universal encoding strategies?, IEEE Trans. Inf. Theory, 52, 12, 5406-5425 (2004) · Zbl 1309.94033 [5] Candes, E. J.; Wakin, M. B., An introduction to compressive sampling, IEEE Signal Process. Mag., 25, 2, 21-30 (2008) [6] Cands; Emmanuel, J., The restricted isometry property and its implications for compressed sensing, C. R. Math., 346, 9, 589-592 (2008) · Zbl 1153.94002 [7] Carrillo, R. E.; Barner, K. E., Lorentzian iterative hard thresholding: robust compressed sensing with prior information, IEEE Trans. Signal Process., 61, 19, 4822-4833 (2013) · Zbl 1393.94188 [8] Carrillo, R. E.; Ramirez, A. B.; Arce, G. R.; Barner, K. E.; Sadler, B. M., Robust compressive sensing of sparse signals: a review, EURASIP J. Adv. Signal Process., 2016, 1, 108 (2016) [9] Chartrand, R.; Yin, W., Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech and Signal Processing, 3869-3872 (2008) [10] Chartrand, R., Exact reconstruction of sparse signals via nonconvex minimization, IEEE Signal Process. Lett., 14, 10, 707-710 (2007) [11] Chen, B.; Xing, L.; Zhao, H.; Zheng, N.; Principe, J. C., Generalized correntropy for robust adaptive filtering, IEEE Trans. Signal Process., 64, 13, 3376-3387 (2016) · Zbl 1414.94113 [12] Chen, B.; Liu, X.; Zhao, H.; Principe, J. C., Maximum correntropy kalman filter, Automatica, 76, 70-77 (2017) · Zbl 1352.93095 [13] Chen, B.; Wang, J.; Zhao, H.; Zheng, N., Convergence of a fixed-point algorithm under maximum correntropy criterion, IEEE Signal Process Lett., 22, 10, 1723-1727 (2015) [14] Chen, B.; Xing, L.; Liang, J.; Zheng, N.; Principe, J. C., Steady-state mean-square error analysis for adaptive filtering under the maximum correntropy criterion, Signal Process. Lett. IEEE, 21, 7, 880-884 (2014) [15] Donoho, D. L., Compressed sensing, IEEE Trans. Inf. Theory, 52, 4, 1289-1306 (2006) · Zbl 1288.94016 [16] Du, B.; Zhang, L.; Tao, D.; Zhang, D., Unsupervised transfer learning for target detection from hyperspectral images, Neurocomputing, 120, 72-82 (2013) [17] Du, B.; Zhang, M.; Zhang, L.; Hu, R.; Tao, D., Pltd: patch-based low-rank tensor decomposition for hyperspectral images, IEEE Trans. Multimed., 19, 1, 67-79 (2017) [18] Elad, M.; Aharon, M., Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15, 12, 3736-3745 (2006) [19] Gan, L., Block compressed sensing of natural images, International Conference on Digital Signal Processing, 403-406 (2007) [20] Gilbert, A.; Indyk, P., Sparse recovery using sparse matrices, Proc. IEEE, 98, 6, 937-947 (2008) [22] He, R.; Wang, L.; Sun, Z.; Zhang, Y.; Li, B., Information theoretic subspace clustering, IEEE Trans. Neural Netw. Learn. Syst., 27, 12, 2643-2655 (2015) [23] Jin, J.; Gu, Y.; Mei, S., A stochastic gradient approach on compressive sensing signal reconstruction based on adaptive filtering framework, IEEE J. Sel. Top. Signal Process., 4, 2, 409-420 (2010) [24] Dennis, J. E., Techniques for nonlinear least squares and robust regression, Commun. Stat.- Simul. Comput., 7, 4, 345-359 (1978) · Zbl 0395.62046 [25] Liu, H.; Liu, Y.; Sun, F., Robust exemplar extraction using structured sparse coding, IEEE Trans. Neural Netw. Learn. Syst., 26, 8, 1816-1821 (2015) [26] Liu, W.; Pokharel, P. P.; Príncipe, J. C., Correntropy: properties and applications in non-gaussian signal processing, Signal Process. IEEE Trans., 55, 11, 5286-5298 (2007) · Zbl 1390.94277 [27] Loza, C. A.; Principe, J. C., Generalized correntropy matching pursuit: a novel, robust algorithm for sparse decomposition, International Joint Conference on Neural Networks, 1723-1727 (2016) [28] Lustig, M.; Donoho, D.; Pauly, J. M., Sparse mri: the application of compressed sensing for rapid mr imaging, Magn. Reson. Med., 58, 6, 1182-1195 (2007) [29] Ma, W.; Qu, H.; Gui, G.; Xu, L.; Zhao, J.; Chen, B., Maximum correntropy criterion based sparse adaptive filtering algorithms for robust channel estimation under non-Gaussian environments, J. Franklin Inst., 352, 7, 2708-2727 (2015) · Zbl 1395.93544 [30] Mallat, S. G.; Zhang, Z., Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Process., 41, 12, 3397-3415 (1993) · Zbl 0842.94004 [31] Ozeki, K.; Umeda, T., An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties, Electron. Commun. Jpn., 67, 5 (1984) [32] Nd, H. K.; Erdogmus, D.; Torkkola, K.; Principe, J. C., Feature extraction using information-theoretic learning, IEEE Trans. Pattern Anal. Mach. Intell., 28, 9 (2006) [33] Ollila, E.; Kim, H. J.; Koivunen, V., Robust iterative hard thresholding for compressed sensing, International Symposium on Communications, Control and Signal Processing, 226-229 (2014) [34] Paredes, J. L.; Arce, G. R., Compressive sensing signal reconstruction by weighted median regression estimates, IEEE Trans. Signal Process., 59, 6, 2585-2601 (2011) · Zbl 1392.94383 [35] Pinkus, A., n-Widths in Approximation Theory (1985), Springer-Verlag · Zbl 0551.41001 [36] Potter, L. C.; Ertin, E.; Parker, J. T.; Cetin, M., Sparsity and compressed sensing in radar imaging, Proc. IEEE, 98, 6, 1006-1020 (2010) [37] Principe, J. C., Information Theoretic Learning: Renyi’s Entropy and Kernel Perspectives (2010), Springer Publishing Company, Incorporated · Zbl 1206.94003 [38] Sayed; Ali, H., Fundamentals of adaptive filtering, Control Syst. IEEE, 25, 4, 77-79 (2003) [39] Singh, A.; Pokharel, R.; Principe, J., The c-loss function for pattern classification, Pattern Recognit., 47, 1, 441-453 (2014) · Zbl 1326.68253 [40] Tibshirani, R., Regression shrinkage and selection via the lasso: a retrospective, J. R. Stat. Soc., 73, 3 (2011) · Zbl 1411.62212 [41] Tropp, J. A.; Gilbert, A. C., Signal recovery from random measurements via orthogonal matching pursuit, Inf. Theory IEEE Trans., 53, 12, 4655-4666 (2007) · Zbl 1288.94022 [42] Tsaig, Y.; Donoho, D. L., Extensions of compressed sensing, Signal Process., 86, 3, 549-571 (2006) · Zbl 1163.94399 [43] Wang, Y.; Yuan, Y. T.; Li, L., Correntropy matching pursuit with application to robust digit and face recognition, IEEE Trans. Cybern., 99, 1-13 (2016) [44] Xiao, Y.; Zhu, H.; Wu, S. Y., Primal and dual alternating direction algorithms for $$l_1 - l_1$$-norm minimization problems in compressive sensing, Comput. Optim. Appl., 54, 2, 441-459 (2013) · Zbl 1269.90081 [45] Yang, J.; Zhang, Y., Alternating direction algorithms for ł; $$_1$$-problems in compressive sensing, Siam J. Sci. Comput., 33, 1, 250-278 (2009) [46] Ye, C.; Gui, G.; Matsushita, S.; Xu, L., Robust stochastic gradient-based adaptive filtering algorithms to realize compressive sensing against impulsive interferences, Control and Decision Conference (CCDC), 2016 Chinese, 1946-1951 (2016), IEEE [47] Zeng, W. J.; So, H. C.; Jiang, X., Outlier-robust greedy pursuit algorithms in $$l_p$$-space for sparse approximation, Signal Process. IEEE Trans., 64, 1, 60-75 (2016) · Zbl 1412.94111 [48] Zhang, Y.; Dong, Z.; Phillips, P.; Wang, S.; Ji, G.; Yang, J., Exponential wavelet iterative shrinkage thresholding algorithm for compressed sensing magnetic resonance imaging, Inf. Sci., 322, 115-132 (2015) [49] Zhou, Y.; Zeng, F., 2D compressive sensing and multi-feature fusion for effective 3d shape retrieval, Inf. Sci., 409, 101-120 (2017) [50] Zou, X.; Feng, L.; Sun, H., Robust compressive sensing of multichannel eeg signals in the presence of impulsive noise, Inf. Sci., 429, 120-129 (2018)
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