@article {JOC_Kernel_RLS_2007,
	title = {Nonlinear System Identification using a New Sliding-Window Kernel {RLS} Algorithm},
	journal = {Journal of Communications},
	volume = {2},
	number = {3},
	year = {2007},
	month = {May},
	pages = {1{\textendash}8},
	abstract = {In this paper we discuss in detail a recently proposed kernel-based version of the recursive least-squares (RLS) algorithm for fast adaptive nonlinear filtering. Unlike other previous approaches, the studied method combines a sliding-window approach (to fix the dimensions of the kernel matrix) with conventional ridge regression (to improve generalization). The resulting kernel RLS algorithm is applied to several nonlinear system identification problems. Experiments show that the proposed algorithm is able to operate in a time-varying environment and to adjust to abrupt changes in either the linear filter or the nonlinearity.},
	keywords = {kernel adaptive filtering, KRLS},
	author = {Van Vaerenbergh, Steven and V{\'\i}a, Javier and Santamar{\'\i}a, Ignacio}
}
@article {282,
	title = {Kernel Recursive Least-Squares Tracker for Time-Varying Regression},
	journal = {IEEE Transactions on Neural Networks and Learning Systems},
	volume = {23},
	year = {2012},
	month = {August},
	pages = {1313--1326},
	abstract = {In this paper, we introduce a kernel recursive least-squares (KRLS) algorithm that is able to track nonlinear, time-varying relationships in data. To this purpose, we first derive the standard KRLS equations from a Bayesian perspective (including a sensible approach to pruning) and then take advantage of this framework to incorporate forgetting in a consistent way, thus enabling the algorithm to perform tracking in nonstationary scenarios. The resulting method is the first kernel adaptive filtering algorithm that includes a forgetting factor in a principled and numerically stable manner. In addition to its tracking ability, it has a number of appealing properties. It is online, requires a fixed amount of memory and computation per time step, incorporates regularization in a natural manner and provides confidence intervals along with each prediction. We include experimental results that support the theory as well as illustrate the efficiency of the proposed algorithm.},
	keywords = {kernel adaptive filtering, KRLS},
	issn = {2162-237X},
	doi = {10.1109/TNNLS.2012.2200500},
	url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6227361},
	author = {Van Vaerenbergh, Steven and L{\'a}zaro-Gredilla, Miguel and Santamar{\'\i}a, Ignacio}
}
@article {342,
	title = {Gaussian Processes for Nonlinear Signal Processing: An Overview of Recent Advances},
	journal = {IEEE Signal Processing Magazine},
	volume = {30},
	year = {2013},
	month = {July},
	pages = {40-50},
	abstract = {Gaussian processes (GPs) are versatile tools that have been successfully employed to solve nonlinear estimation problems in machine learning, but that are rarely used in signal processing. In this tutorial, we present GPs for regression as a natural nonlinear extension to optimal Wiener filtering. After establishing their basic formulation, we discuss several important aspects and extensions, including recursive and adaptive algorithms for dealing with non-stationarity, low-complexity solutions, non-Gaussian noise models and classification scenarios. Furthermore, we provide a selection of relevant applications to wireless digital communications.},
	keywords = {kernel adaptive filtering},
	issn = {1053-5888},
	doi = {10.1109/MSP.2013.2250352},
	url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6530761},
	author = {P{\'e}rez-Cruz, Fernando and Van Vaerenbergh, Steven and Murillo-Fuentes, Juan Jos{\'e} and L{\'a}zaro-Gredilla, Miguel and Santamar{\'\i}a, Ignacio}
}
@conference {vanvaerenbergh2006sliding,
	title = {A Sliding-Window Kernel {RLS} Algorithm and its Application to Nonlinear Channel Identification},
	booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2006)},
	year = {2006},
	month = {May},
	address = {Toulouse, France},
	abstract = {In this paper we propose a new kernel-based version of the recursive least-squares (RLS) algorithm for fast adaptive nonlinear filtering. Unlike other previous approaches, we combine a sliding-window approach (to fix the dimensions of the kernel matrix) with conventional -norm regularization (to improve generalization). The proposed kernel RLS algorithm is applied to a nonlinear channel identification problem (specifically, a linear filter followed by a memoryless nonlinearity), which typically appears in satellite communications or digital magnetic recording systems. We show that the proposed algorithm is able to operate in a time-varying environment and tracks abrupt changes in either the linear filter or the nonlinearity.},
	keywords = {kernel adaptive filtering, KRLS},
	author = {Van Vaerenbergh, Steven and V{\'\i}a, Javier and Santamar{\'\i}a, Ignacio}
}
@conference {471,
	title = {On The Relationship Between Online Gaussian Process Regression And Kernel Least Mean Squares Algorithms},
	booktitle = {2016 IEEE International Workshop on Machine Learning for Signal Processing (MLSP 2016)},
	year = {2016},
	month = {September},
	publisher = {IEEE},
	organization = {IEEE},
	address = {Salerno, Italy},
	abstract = {We study the relationship between online Gaussian process (GP) regression and kernel least mean squares (KLMS) algorithms. While the latter have no capacity of storing the entire posterior distribution during online learning, we discover that their operation corresponds to the assumption of a fixed posterior covariance that follows a simple parametric model. Interestingly, several well-known KLMS algorithms correspond to specific cases of this model. The probabilistic perspective allows us to understand how each of them handles uncertainty, which could explain some of their performance differences.},
	keywords = {gaussian processes, kernel adaptive filtering, KLMS},
	author = {Van Vaerenbergh, Steven and Fern{\'a}ndez-Bes, Jes{\'u}s and Elvira, Victor}
}
@conference {393,
	title = {Probabilistic Kernel Least Mean Squares Algorithms},
	booktitle = {2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
	year = {2014},
	month = {05/2014},
	abstract = {The kernel least mean squares (KLMS) algorithm is a computationally efficient nonlinear adaptive filtering method that "kernelizes" the celebrated (linear) least mean squares algorithm. We demonstrate that the least mean squares algorithm is closely related to the Kalman filtering, and thus, the KLMS can be interpreted as an approximate Bayesian filtering method. This allows us to systematically develop extensions of the KLMS by modifying the underlying state-space and observation models. The resulting extensions introduce many desirable properties such as "forgetting", and the ability to learn from discrete data, while retaining the computational simplicity and time complexity of the original algorithm.},
	keywords = {kernel adaptive filtering, KLMS, sequential Bayesian learning, state-space model},
	isbn = {978-1-4799-2892-7},
	author = {Park, Il Memming and Seth, Sohan and Van Vaerenbergh, Steven}
}
@conference {362,
	title = {Kernel Adaptive Filtering: Which Technique to Choose in Practice},
	booktitle = {International Workshop on Advances in  Regularization, Optimization, Kernel Methods and Support Vector Machines: theory and applications (ROKS 2013)},
	year = {2013},
	month = {July},
	pages = {101--102},
	abstract = {The field of kernel adaptive filtering has produced a myriad of techniques throughout the past decade. While each algorithm provides some advantages over others in certain scenarios, it is often not clear which technique should be used on a practical problem, in which specific restrictions are in place. We propose a set of empirical performance measures and we provide a framework that can be used to decide which algorithm to use in practice. We include results on several benchmark data sets.},
	keywords = {kernel adaptive filtering},
	isbn = {978-94-6018-700-1},
	author = {Van Vaerenbergh, Steven and Santamar{\'\i}a, Ignacio}
}
@conference {vanvaerenbergh2010_fbkrls,
	title = {Fixed-Budget Kernel Recursive Least-Squares},
	booktitle = {IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2010)},
	year = {2010},
	month = {March},
	address = {Dallas, USA},
	keywords = {kernel adaptive filtering, KRLS},
	author = {Van Vaerenbergh, Steven and Santamar{\'\i}a, Ignacio and Liu, Weifeng and Pr{\'\i}ncipe, Jos{\'e} C.}
}
@conference {314,
	title = {Estimation of the Forgetting Factor in Kernel Recursive Least Squares},
	booktitle = {2012 IEEE International Workshop On Machine Learning For Signal Processing (MLSP)},
	year = {2012},
	month = {September},
	abstract = {In a recent work we proposed a kernel recursive least-squares tracker (KRLS-T) algorithm that is capable of tracking in non-stationary environments, thanks to a forgetting mechanism built on a Bayesian framework. In order to guarantee optimal performance its parameters need to be determined, specifically its kernel parameters, regularization and, most importantly in non-stationary environments, its forgetting factor. This is a common difficulty in adaptive filtering techniques and in signal processing algorithms in general. In this paper we demonstrate the equivalence between KRLS-T{\textquoteright}s recursive tracking solution and Gaussian process (GP) regression with a specific class of spatio-temporal covariance. This result allows to use standard hyperparameter estimation techniques from the Gaussian process framework to determine the parameters of the KRLS-T algorithm. Most notably, it allows to estimate the optimal forgetting factor in a principled manner. We include results on different benchmark data sets that offer interesting new insights.},
	keywords = {kernel adaptive filtering},
	author = {Van Vaerenbergh, Steven and Santamar{\'\i}a, Ignacio and L{\'a}zaro-Gredilla, Miguel}
}
@conference {356,
	title = {A Comparative Study of Kernel Adaptive Filtering Algorithms},
	booktitle = {2013 IEEE Digital Signal Processing (DSP) Workshop and IEEE Signal Processing Education (SPE)},
	year = {2013},
	month = {August},
	publisher = {IEEE},
	organization = {IEEE},
	abstract = {Kernel adaptive filtering is a growing field of signal processing that is concerned with nonlinear adaptive filtering. When implemented naively, the time and memory complexities of these algorithms grow at least linearly with the amount of data processed. A large number of practical solutions have been proposed throughout the last decade, based on sparsification or pruning mechanisms. Nevertheless, there is a lack of understanding of their relative merits, which often depend on the data they operate on. We propose to study the quality of the solution as a function of either the time or the memory complexity. We empirically test six different kernel adaptive filtering algorithms on three different benchmark data sets. We make our code available through an open source toolbox that includes additional algorithms and allows to measure the complexities explicitly in number of floating point operations and bytes needed, respectively.},
	keywords = {kernel adaptive filtering},
	isbn = {978-1-4799-1616-0},
	doi = {10.1109/DSP-SPE.2013.6642587 },
	author = {Van Vaerenbergh, Steven and Santamar{\'\i}a, Ignacio}
}
@conference {Miguel_KRLS_MLSP2011,
	title = {A {Bayesian} Approach To Tracking With Kernel Recursive Least-Squares},
	booktitle = {IEEE International Workshop on Machine Learning for Signal Processing (MLSP 2011)},
	year = {2011},
	month = {September},
	address = {Beijing, China},
	abstract = {In this paper we introduce a kernel-based recursive least-squares (KRLS) algorithm that is able to track nonlinear, time-varying relationships in data. To this purpose we first derive the standard KRLS equations from a Bayesian perspective (including a principled approach to pruning) and then take advantage of this framework to incorporate forgetting in a consistent way, thus enabling the algorithm to perform tracking in non-stationary scenarios. In addition to this tracking ability, the resulting algorithm has a number of appealing properties: It is online, requires a fixed amount of memory and computation per time step and incorporates regularization in a natural manner. We include experimental results that support the theory as well as illustrate the efficiency of the proposed algorithm.},
	keywords = {kernel adaptive filtering, KRLS},
	author = {L{\'a}zaro-Gredilla, Miguel and Van Vaerenbergh, Steven and Santamar{\'\i}a, Ignacio}
}
