Bump hunting with non-Gaussian kernels

Chunming Zhang; Michael C. Minnotte; Peter Hall

doi:10.1214/009053604000000715

Bump hunting with non-Gaussian kernels

The Annals of Statistics ◽

10.1214/009053604000000715 ◽

2004 ◽

Vol 32 (5) ◽

pp. 2124-2141 ◽

Cited By ~ 24

Author(s):

Chunming Zhang ◽

Michael C. Minnotte ◽

Peter Hall

Keyword(s):

Gaussian Kernels ◽

Bump Hunting ◽

Non Gaussian

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Optimal learning with Gaussians and correntropy loss

Analysis and Applications ◽

10.1142/s0219530519410124 ◽

2019 ◽

Vol 19 (01) ◽

pp. 107-124

Author(s):

Fusheng Lv ◽

Jun Fan

Keyword(s):

Least Squares Method ◽

Approximation Error ◽

Regression Function ◽

Polynomial Decay ◽

Gaussian Kernel ◽

Smoothness Condition ◽

Great Success ◽

Information Theoretic Learning ◽

Gaussian Kernels ◽

Non Gaussian

Correntropy-based learning has achieved great success in practice during the last decades. It is originated from information-theoretic learning and provides an alternative to classical least squares method in the presence of non-Gaussian noise. In this paper, we investigate the theoretical properties of learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and correntropy loss. By choosing an appropriate scale parameter of Gaussian kernel, we show the polynomial decay of approximation error under a Sobolev smoothness condition. In addition, we employ a tight upper bound for the uniform covering number of Gaussian RKHS in order to improve the estimate of sample error. Based on these two results, we show that the proposed algorithm using varying Gaussian kernel achieves the minimax rate of convergence (up to a logarithmic factor) without knowing the smoothness level of the regression function.

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Enhanced Data Covariance Estimation Using Weighted Combination of Multiple Gaussian Kernels for Improved M/EEG Source Localization

International Journal of Neural Systems ◽

10.1142/s0129065719500011 ◽

2019 ◽

Vol 29 (06) ◽

pp. 1950001

Author(s):

J. D. Martinez-Vargas ◽

L. Duque-Muñoz ◽

F. Vargas-Bonilla ◽

J. D. Lopez ◽

G. Castellanos-Dominguez

Keyword(s):

Brain Activity ◽

Covariance Estimation ◽

Open Problems ◽

Noninvasive Technique ◽

Gaussian Kernels ◽

Brain Functions ◽

Eeg Data ◽

Leibler Divergence ◽

Non Gaussian ◽

Weighted Combination

In the recent past, estimating brain activity with magneto/electroencephalography (M/EEG) has been increasingly employed as a noninvasive technique for understanding the brain functions and neural dynamics. However, one of the main open problems when dealing with M/EEG data is its non-Gaussian and nonstationary structure. In this paper, we introduce a methodology for enhancing the data covariance estimation using a weighted combination of multiple Gaussian kernels, termed WM-MK, that relies on the Kullback–Leibler divergence for associating each kernel weight to its relevance. From the obtained results of validation on nonstationary and non-Gaussian brain activity (simulated and real-world EEG data), WM-MK proves that the accuracy of the source estimation raises by more effectively exploiting the measured nonlinear structures with high time and space complexity.

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