Complex kernels for proper complex-valued signals: A review

Boloix-Tortosa, Rafael; Payán-Somet, F. Javier; Arias-de-Reyna, Eva; Murillo-Fuentes, Juan José

doi:10.1109/eusipco.2015.7362809

Cited by 8 publications

(5 citation statements)

References 11 publications

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“…However, this kernel measures similarities between the real parts of the inputs with |x r − x r | 2 , while for the imaginary ones it uses |x j + x j | 2 , where | · | is the 2 -norm. Also, it is not stationary, has an oscillatory behavior, and the exponent in the kernel may easily grow large and positive [26]. This might cause numerical problems and, as we show later in the experiments, it does not yield the best performance.…”

Section: A Kernel Designmentioning

confidence: 92%

“…. Also, it is not stationary, has an oscillatory behavior, and the exponent in the kernel may easily grow large and positive [26]. This might cause numerical problems and, as we show later in the experiments, it does not yield the best performance.…”

Section: A Kernel Designmentioning

confidence: 92%

“…We will use it in our experiments. For a further analysis about the selection of suitable kernels for complex-valued applications see [11], [26].…”

Section: A Kernel Designmentioning

confidence: 99%

See 2 more Smart Citations

The Generalized Complex Kernel Least-Mean-Square Algorithm

Boloix-Tortosa

Murillo-Fuentes

Tsaftaris

2019

IEEE Trans. Signal Process.

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We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complexvalued kernel least-mean-square (gCKLMS). We borrow from the new results on widely linear reproducing kernel Hilbert space (WL-RKHS) for nonlinear regression and complex-valued signals, recently proposed by the authors. This paper shows that in the adaptive version of the kernel regression for complexvalued signals we need to include another kernel term, the socalled pseudo-kernel. This new solution is endowed with better representation capabilities in complex-valued fields, since it can efficiently decouple the learning of the real and the imaginary part. Also, we review previous realizations of the complex KLMS algorithm and its augmented version to prove that they can be rewritten as particular cases of the gCKLMS. Furthermore, important conclusions on the kernels design are drawn that help to greatly improve the convergence of the algorithms. In the experiments, we revisit the nonlinear channel equalization problem to highlight the better convergence of the gCKLMS compared to previous solutions. Also, the flexibility of the proposed generalized approach is tested in a second experiment with non-independent real and imaginary parts. The results illustrate the significant performance improvements of the gCKLMS approach when the complex-valued signals have different properties for the real and imaginary parts.

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Section: A Kernel Designmentioning

confidence: 92%

Section: A Kernel Designmentioning

confidence: 92%

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The Generalized Complex Kernel Least-Mean-Square Algorithm

Boloix-Tortosa

Murillo-Fuentes

Tsaftaris

2019

IEEE Trans. Signal Process.

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“…The kernel matrix K = [K ij := κ( i , j )] can be written concisely as K := Φ(Λ) H Φ(Λ). A range of kernel functions is available for complex RKHS, such as the Gaussian κ( i , j ) = exp(−γ i − * j 2 ) ( * represents vector/matrix conjugation) and the polynomial kernel κ( i , j ) = ( i H j + c) r [23]. The following assumption imposes a structure on φ(y nav j ); a condition often met in manifold-learning approaches [24].…”

Section: Data Modelingmentioning

confidence: 99%

Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery

Shetty

Slavakis

Bose

et al. 2020

IEEE Trans. Med. Imaging

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This paper establishes a kernel-based framework for reconstructing data on manifolds, tailored to fit the dynamic-(d)MRIdata recovery problem. The proposed methodology exploits simple tangent-space geometries of manifolds in reproducing kernel Hilbert spaces, and follows classical kernel-approximation arguments to form the data-recovery task as a bi-linear inverse problem. Departing from mainstream approaches, the proposed methodology uses no training data, employs no graph Laplacian matrix to penalize the optimization task, uses no costly (kernel) preimaging step to map feature points back to the input space, and utilizes complex-valued kernel functions to account for k-space data. The framework is validated on synthetically generated dMRI data, where comparisons against state-of-the-art schemes highlight the rich potential of the proposed approach in data-recovery problems.

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“…While the more common smooth squared exponential (SE) kernel is used in this article, the Matern class of kernels could be used for systems with sharper features in the frequency response, e.g., for underdamped systems 33) and the approach could be applied to complex-valued kernels in the frequency domain. 36) The concept of model update in the frequency domain proposed here could be used with spatial-domain iterations, e.g., 37), 38) with model identification methods using repetitive trajectories, 39) and with the stable spline kernel for machine learning in the time domain 35) that guarantees bounded-inputbounded-output stability of the resulting models. Finally, the proposed approach can be used to speed up the learning of the different segments in segmented iterative control approaches, e.g.,.…”

Section: )-18)mentioning

confidence: 99%

Iterative Machine Learning for Output Tracking

Devasia

2019

IEEE Trans. Contr. Syst. Technol.

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This article develops iterative machine learning (IML) for output tracking. The inputoutput data generated during iterations to develop the model used in the iterative update. The main contribution of this article to propose the use of kernel-based machine learning to iteratively update both the model and the model-inversion-based input simultaneously. Additionally, augmented inputs with persistency of excitation are proposed to promote learning of the model during the iteration process. The proposed approach is illustrated with a simulation example. §1. IntroductionIterative learning methods, initially developed in e.g.,, 1)-3) improve the outputtracking performance by correcting the input based on the measured tracking error. For example, iterative control has led to some of the highest precision for output tracking, e.g., in scanning probe microscopy as demonstrated in, e.g.,. 4)-10) Note that sets of learned trajectories can be used to enable tracking of other trajectories, e.g., by designing the feedforward control input using pre-specified basis functions e.g., using polynomial functions 11) or rational functions of the reference trajectory. 12) Similarly, new desired output can be generated by considering different combinations of previously-learned output segments 13) and the feedforward can be represented using radial basis functions that can be optimized for a range of task parameters. 14) This article proposes a kernel-based machine learning approach to use augmented inputs to iteratively learn not just the inverse input u d needed to track a specified output y d but to also use the data acquired during the iteration process to to estimate both (i) the modelĝ (and its inverse) for the control update, as well as (ii) the model uncertainty needed to establish bounds on the iteration gain for ensuring trackingerror reduction..Conditions for convergence of iterative methods have been well studied in literature, e.g.,. 15)-24) For example, the need to invert the system g to find the perfect input u d for tracking a desired output y d has motivated the use of the inversê g −1 of the known modelĝ of the system g in early iterative control development. 3) Since convergence depends on the size of modeling error, improvements of the model through parameter adaptation with data acquired during the iteration was studied in, e.g., 25) for robotics application using a discrete-time implementation. Here, for each iteration step, the sampled input vector is mapped to the sampled output vector through a lower triangular matrix map. A stochastic version using such a lower-triangular map has been studied in. 26) Even in the ideal case with no mod- * ) S. Devasia is with the Mechanical Engineering Department, U.

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Complex kernels for proper complex-valued signals: A review

Cited by 8 publications

References 11 publications

The Generalized Complex Kernel Least-Mean-Square Algorithm

The Generalized Complex Kernel Least-Mean-Square Algorithm

Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery

Iterative Machine Learning for Output Tracking

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