{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T06:36:19Z","timestamp":1762324579532,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,22]],"date-time":"2019-10-22T00:00:00Z","timestamp":1571702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61671389"],"award-info":[{"award-number":["61671389"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["XDJK2019B011"],"award-info":[{"award-number":["XDJK2019B011"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Random Fourier mapping (RFM) in kernel adaptive filters (KAFs) provides an efficient method to curb the linear growth of the dictionary by projecting the original input data into a finite-dimensional space. The commonly used measure in RFM-based KAFs is the minimum mean square error (MMSE), which causes performance deterioration in the presence of non-Gaussian noises. To address this issue, the minimum Cauchy loss (MCL) criterion has been successfully applied for combating non-Gaussian noises in KAFs. However, these KAFs using the well-known stochastic gradient descent (SGD) optimization method may suffer from slow convergence rate and low filtering accuracy. To this end, we propose a novel robust random Fourier features Cauchy conjugate gradient (RFFCCG) algorithm using the conjugate gradient (CG) optimization method in this paper. The proposed RFFCCG algorithm with low complexity can achieve better filtering performance than the KAFs with sparsification, such as the kernel recursive maximum correntropy algorithm with novelty criterion (KRMC-NC), in stationary and non-stationary environments. Monte Carlo simulations conducted in the time-series prediction and nonlinear system identification confirm the superiorities of the proposed algorithm.<\/jats:p>","DOI":"10.3390\/sym11101323","type":"journal-article","created":{"date-parts":[[2019,10,23]],"date-time":"2019-10-23T11:46:59Z","timestamp":1571831219000},"page":"1323","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Cauchy Conjugate Gradient Algorithm with Random Fourier Features"],"prefix":"10.3390","volume":"11","author":[{"given":"Xuewei","family":"Huang","sequence":"first","affiliation":[{"name":"College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China"},{"name":"Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing 400715, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5028-5839","authenticated-orcid":false,"given":"Shiyuan","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China"},{"name":"Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing 400715, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kui","family":"Xiong","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China"},{"name":"Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing 400715, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1540","DOI":"10.1109\/TSP.2004.830991","article-title":"Online learning with kernels","volume":"52","author":"Kivinen","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"doi-asserted-by":"crossref","unstructured":"Liu, W., Pr\u00edncipe, J.C., and Haykin, S. (2010). Kernel Adaptive Filtering: A Comprehensive Introduction, John Wiley & Sons.","key":"ref_2","DOI":"10.1002\/9780470608593"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1109\/TSP.2007.907881","article-title":"The kernel least mean square algorithm","volume":"56","author":"Liu","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2008\/784292","article-title":"Kernel affine projection algorithms","volume":"2008","author":"Liu","year":"2008","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2275","DOI":"10.1109\/TSP.2004.830985","article-title":"The kernel recursive least squares algorithm","volume":"52","author":"Engel","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1950","DOI":"10.1109\/TNET.2012.2187923","article-title":"An information theoretic approach of designing sparse kernel adaptive filters","volume":"20","author":"Liu","year":"2009","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1058","DOI":"10.1109\/TSP.2008.2009895","article-title":"Online prediction of time series data with kernels","volume":"57","author":"Richard","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.dsp.2017.01.010","article-title":"Quantized kernel maximum correntropy and its mean square convergence analysis","volume":"63","author":"Wang","year":"2017","journal-title":"Dig. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2759","DOI":"10.1016\/j.sigpro.2013.02.012","article-title":"Fixed budget quantized kernel least mean square algorithm","volume":"93","author":"Zhao","year":"2013","journal-title":"Signal Process."},{"unstructured":"Vaerenbergh, S.V., V\u00eda, J., and Santamar\u00eda, I. (2006, January 14\u201319). A sliding-window kernel RLS algorithm and its application to nonlinear channel identification. Proceedings of the 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings (ICASSP), Toulouse, France.","key":"ref_10"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3133","DOI":"10.1007\/s00034-018-1006-2","article-title":"Kernel least mean square based on the Nystr\u00f6m method","volume":"38","author":"Wang","year":"2019","journal-title":"Circuits Syst. Signal Process."},{"unstructured":"Rahimi, A., and Recht, B. (2007, January 3\u20136). Random features for large-scale kernel machines. Proceedings of the 21th Annual Conference on Neural Information Processing Systems (ACNIPS), Vancouver, BC, Canada.","key":"ref_12"},{"doi-asserted-by":"crossref","unstructured":"Singh, A., Ahuja, N., and Moulin, P. (2012, January 23\u201326). Online learning with kernels: Overcoming the growing sum problem. Proceedings of the 2012 IEEE International Workshop on Machine Learning for Signal Process (MLSP), Santander, Spain.","key":"ref_13","DOI":"10.1109\/MLSP.2012.6349811"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3390","DOI":"10.1109\/TCSI.2018.2825241","article-title":"Random Fourier filters under maximum correntropy criterion","volume":"65","author":"Wang","year":"2018","journal-title":"IEEE Trans. Circuits Syst. I Reg. Pap."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"740","DOI":"10.1109\/LSP.2019.2907480","article-title":"The online random Fourier features conjugate gradient algorithm","volume":"26","author":"Xiong","year":"2019","journal-title":"IEEE Signal Process. Lett."},{"doi-asserted-by":"crossref","unstructured":"Wu, Q., Li, Y., and Xue, W. (2019). A kernel recursive maximum versoria-like criterion algorithm for nonlinear channel equalization. Symmetry, 11.","key":"ref_16","DOI":"10.3390\/sym11091067"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1109\/TASSP.1987.1165167","article-title":"Improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm","volume":"35","author":"Mathews","year":"1987","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1109\/TIT.1984.1056886","article-title":"The least mean fourth (LMF) adaptive algorithm and its family","volume":"42","author":"Walach","year":"1984","journal-title":"IEEE Trans. Inf. Theory"},{"doi-asserted-by":"crossref","unstructured":"Pr\u00edncipe, J.C. (2010). Information Theoretic Learning: Renyi\u2019s Entropy and Kernel Perspectives, Springer.","key":"ref_19","DOI":"10.1007\/978-1-4419-1570-2"},{"doi-asserted-by":"crossref","unstructured":"Li, Y., Wang, Y., and Sun, L. (2018). A proportionate normalized maximum correntropy criterion algorithm with correntropy induced metric constraint for identifying sparse systems. Symmetry, 10.","key":"ref_20","DOI":"10.3390\/sym10120683"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1080\/00949655.2013.874424","article-title":"A cauchy estimator test for autocorrelation","volume":"85","author":"Gallagher","year":"2015","journal-title":"J. Stat. Comput. Simul."},{"doi-asserted-by":"crossref","unstructured":"Luenberger, D.G. (2016). Linear and Nonlinear Programming, Prentice Hall. [4th ed.].","key":"ref_22","DOI":"10.1007\/978-3-319-18842-3"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.neunet.2017.04.001","article-title":"Recursive least mean p-Power extreme learning machine","volume":"91","author":"Yang","year":"2017","journal-title":"Neural Netw."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/81.109237","article-title":"Conjugate gradient techniques for adaptive filtering","volume":"39","author":"Boray","year":"1992","journal-title":"IEEE Trans. Circuits Syst. I Fundam. Theory Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1109\/78.823968","article-title":"Analysis of conjugate gradient algorithms for adaptive filtering","volume":"48","author":"Chang","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"409","DOI":"10.6028\/jres.049.044","article-title":"Methods of conjugate gradients for solving linear systems","volume":"49","author":"Hestenes","year":"1952","journal-title":"J. Res. Nat. Bur. Stand."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2783","DOI":"10.1137\/141002062","article-title":"A preconditioner for a primal-dual newton conjugate gradients method for compressed sensing problems","volume":"37","author":"Dassios","year":"2015","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6252","DOI":"10.1109\/TNNLS.2018.2827778","article-title":"A new correntropy-based conjugate gradient backpropagation algorithm for improving training in neural networks","volume":"29","author":"Heravi","year":"2018","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_29","first-page":"196","article-title":"Preconditioned nonlinear conjugate gradient methods based on a modified secant equation","volume":"318","author":"Caliciotti","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4377","DOI":"10.1109\/TSP.2018.2853109","article-title":"The kernel conjugate gradient algorithms","volume":"66","author":"Zhang","year":"2018","journal-title":"Trans. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.sigpro.2015.04.024","article-title":"Kernel recursive maximum correntropy","volume":"117","author":"Wu","year":"2015","journal-title":"Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1109\/TNNLS.2013.2258936","article-title":"Quantized kernel recursive least squares algorithm","volume":"24","author":"Chen","year":"2013","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3376","DOI":"10.1109\/TSP.2016.2539127","article-title":"Generalized correntropy for robust adaptive filtering","volume":"64","author":"Chen","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2588","DOI":"10.1109\/TSP.2005.849213","article-title":"Nonlinear system identification in impulsive environments","volume":"53","author":"Weng","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1080\/00207179208934272","article-title":"Recursive hybrid algorithm for non-linear system identification using radial basis function networks","volume":"55","author":"Chen","year":"1992","journal-title":"Int. J. 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