Overview

Gaussian processes are non-parametric kernel based Bayesian tools to perform inference. Non-parametric kernel solutions are based on providing a new solution for some new input by using the set of training data. This set is included in the so-called kernel matrix.

Gaussian processes for regression (GPR) are useful tool to perform prediction or even detection. It exhibits three major advantages. First, the formulation is analytic and so it is its solution. This solution can be cast as a “non-linear” version of the linear minimum mean square error (MMSE) estimator with extra regularization term. Second, the hyper-parameter learning can be performed in a principled way, rather than using cross-validation. Third, it does not only provides a predicted or estimated value, but a probabilistic information on it. Its counterpart for classification, GP for classification (GPC), exhibits the same features but that of being analytic.

Since MMSE has been a quite extended tool in digital communications, we forecasted that GPR and GPC would be useful there were the linear MMSE failed. Both, in the inversion of linear and non-linear systems. We proved its good features in the multiuser detection and channel equalization. Furthermore, we exploited its output being probabilistic to feed them to modern channel decoders, quite improving the overall performance.

In some other applications we found it a competitive tool. As in the prediction of tax incomes in Brazil, where we collaborated with the University of Brazilia.

Lately, and given the interest to develop non-parametric tools for complex-valued signals, we are working on the formulation of GPR for both, proper and non-proper complex signals. With application, among others, to channel equalization.

Involved Members

Juan J. Murillo-Fuentes, Fernando Pérez-Cruz, Rafael Boloix-Tortosa, Pablo M. Olmos, Javier Payán-Somet, Eva Arias de Reyna.

Publications

Journals

  1. R. Boloix-Tortosa, J.J. Murillo-Fuentes S.A. Tsaftaris. (2019). “The Generalized Complex Kernel Least-Mean-Square Algorithm” IEEE Transactions on Signal Processing, Vol. 67, no. 20, pp. 5213 – 5222. Oct.15, 2019. DOI 10.1109/TSP.2019.2937289, https://arxiv.org/abs/1902.08692
  2. Santos, J.J. Murillo-Fuentes, P. M. Djuric. (2019). Recursive Estimation of Dynamic RSS Fields Based on Crowdsourcing and Gaussian Processes. IEEE Transactions on Signal Processing. 2019. EEE (2017) Q1, 32/260. DOI 10.1109/TSP.2018.2889987, https://arxiv.org/abs/1806.02530.
  3. R. Boloix-Tortosa, J. J. Murillo-Fuentes, F. J. Payán-Somet and F. Pérez-Cruz, “Complex Gaussian Processes for Regression,” in IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 11, pp. 5499-5511, Nov. 2018. DOI: 10.1109/TNNLS.2018.2805019 https://arxiv.org/abs/1511.05710
  4. R. Boloix-Tortosa, J. J. Murillo-Fuentes, I. Santos and F. Pérez-Cruz, (2017) “Widely Linear Complex-Valued Kernel Methods for Regression,” in IEEE Transactions on Signal Processing, vol. 65, no. 19, pp. 5240-5248, Oct.1, 1 2017. Paper, also in IEEE TSP, arxiv.org
  5. F. Pérez-Cruz, S. Van Vaerenbergh, J. J. Murillo-Fuentes, M. Lázaro-Gredilla and I. Santamaría. ”Gaussian Processes for Nonlinear Signal Processing”. IEEE Signal Processing Magazine. Vol.30, no.4. 2013.
  6. P. M. Olmos, J. J. Murillo-Fuentes and F. Pérez-Cruz, (2010). ”Joint Nonlinear Channel Equalization and Soft LDPC Decoding with Gaussian Processes”. IEEE Transactions on Signal Processing, Vol. 58, N. 3, pp. 1183 – 1192, March 2010.
  7. J. J. Murillo-Fuentes and F. Pérez-Cruz, (2009). ”Gaussian Process Regressors for Multiuser Detection in DS-CDMA Systems”. IEEE Transactions on Communications,
    57(8):2339-2347, August 2009.
  8. F. Pérez-Cruz, J. J. Murillo-Fuentes and S. Caro, (2008). ”Nonlinear Channel Equalization with Gaussian Processes for Regression”. IEEE Transactions on Signal Processing, 56(10-2):5283-5286, October 2008.
  9. F. Pérez-Cruz and J. J. Murillo-Fuentes, (2008). ”Digital Communication Receivers Using Gaussian Processes for Machine Learning”. Journal on Advances in Signal Processing. Vol 2008. doi:10.1155/2008/491503.

Book Chapters

Conferences

  1. Havasi, J.M. Hernández-Lobato, J.J. Murillo-Fuentes “Inference in Deep Gaussian Processes using Stochastic Gradient Hamiltonian Monte Carlo“ Neural Information and Processing Systems (NeurIPS), 2018.
  2. I. Santos, P. Djuric, “Crowdsource-based signal strength field estimation by Gaussian Processes”. Proc of 25th European Signal Processing Conference (EUSIPCO). Kos (Greece). 2017
  3. I. Santos Velázquez, J.J. Murillo-Fuentes, P.M. Djuric, (2017) “Recursive Estimation of Time-Varying RSS Fields Based on Crowdsourcing and Gaussian Processes”. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) 2017.
  4. R. Boloix-Tortosa, F. J. Payán-Somet, E. Arias-de-Reyna and J. J. Murillo-Fuentes, (2015) “Complex kernels for proper complex-valued signals: A review,” 23rd European Signal Processing Conference (EUSIPCO), Nice, 2015, pp. 2371-2375. ieeexplore
  5. R. Boloix-Tortosa, F.J. Payán-Somet, J.J. Murillo-Fuentes (2014). ”Gaussian processes regressors for complex proper signals in digital communications”. IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM),  pp. 137-140, 22-25 June 2014.
  6. Pablo M. Olmos, Juan José Murillo-Fuentes and Fernando Pérez-Cruz (2009).”Soft LDPC decoding in nonlinear channels with Gaussian processes for classification”. European Signal Processing Conference (EUSIPCO). Pp. 1641-1645. Glasgow, UK. 24-28 Aug 2009.
  7. Sebastián Caro, Juan José Murillo-Fuentes, Fernando Pérez-Cruz (2006). ”Gaussian Processes for Regression In Channel Equalization”. European Signal Processing Conference (EUSIPCO). Florence, Italy. 4-8 Sep 2006.
  8. Fernando Pérez-Cruz, Juan José Murillo-Fuentes (2006) ”Gaussian Processes for Digital Communications”. In Proc of the IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), Vol V. PP 781-784. ISSN: 1520-6149. ISBN: 1-4244-0469-X. Toulouse (France), 14-19 May 2006.
  9. Fernando Pérez-Cruz, Sebastián Caro, Juan José Murillo-Fuentes (2005). ”Gaussian Processes for Multiuser Detection in CDMA receivers”. Advances in Neural Information Processing Systems 18 (NIPS). Editores Y. Weiss, B. Schölkopf and J. Platt. MIT Press (Cambridge, MA). Pp. 939-946, Vancouver (Canada), 6-9 Dec 2005.

Acknowledgements

These results were possible thanks to public fundings. The Universidad de Sevilla trusted us to carry out this research. The Spanish Government and the European Union (FEDER) also founded this research through the projects MEC.CICYT.TIC 2003-03781, TEC2006-13514-C02-2/TCM, CONSOLIDER CSD2008-00010 and TEC2012-38800-C03-C02. This work was also possible thanks to the fruitful collaboration with prof. Fernando Pérez-Cruz, from Universidad Carlos III de Madrid, member of the group.

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