@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}
}
@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 {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 {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}
}
@inbook {423,
	title = {Online Regression with Kernels},
	booktitle = {Regularization, Optimization, Kernels, and Support Vector Machines},
	number = {Machine Learning \& Pattern Recognition Series},
	year = {2014},
	pages = {477-501},
	publisher = {Chapman and Hall/CRC},
	organization = {Chapman and Hall/CRC},
	chapter = {21},
	address = {New York},
	abstract = {Online machine learning algorithms are designed to learn from one data instance at a time. They are typically used in real-time scenarios, such as prediction or tracking problems, where data arrive sequentially and instant decisions must be made. The real-time nature of these settings implies that shortly after the decision is made, the true label will be made available, which allows the learning algorithm to adjust its solution before a new datum is received.

Online kernel methods extend the nonlinear learning capabilities of standard batch kernel methods to online environments. Especially important for these techniques is that they maintain their computational load moderate during each iteration, in order to perform fast updates in real time. Ideally, they should not only be able to learn in a stationary environment but also in non-stationary settings, where they must forget outdated information and adapt their solution to respond to changes in time. Online kernel methods also find use in batch scenarios where the amount of data is too high to fit in the machine{\textquoteright}s memory, and one or several passes over the data are to be performed.

In this chapter we focus on the problem of online regression. We will give an overview of the most important kernel-based methods for this problem, which have been developed over the span of the last decade. We start by formulating the online solution to the kernel ridge regression problem, and we point out different strategies to overcome the bottlenecks associated to using kernels in online methods. The discussed techniques are often referred to as kernel adaptive filtering algorithms, due to their close relationship with classical adaptive filters from the signal processing literature. After reviewing the most relevant algorithms in this area, we introduce an evaluation framework that allows us to compare their performance. We finish the discussion with a brief overview of the recent and future research directions.},
	keywords = {KLMS, KRLS},
	isbn = {9781482241396},
	url = {http://www.crcpress.com/product/isbn/9781482241396},
	author = {Van Vaerenbergh, Steven and Santamar{\'\i}a, Ignacio}
}
