A data-driven bandwidth choice for a kernel density estimator called critical bandwidth is investigated. This procedure allows the estimation to have as many modes as assumed for the density to estimate. Both Gaussian and uniform kernels are considered. For the Gaussian kernel, asymptotic results are given. For the uniform kernel, an argument against these properties is mentioned. These theoret...

For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an n× n positive definite matrix, and its derivatives – leading to prohibitive O(n) computations. We propose novel O(n) approaches to estimating these quantities from only fast matrix vector mu...

An algorithm of Dutt and Rokhlin (SIAM J Sci Comput 1993;14:1368-1383) for the computation of a fast Fourier transform (FFT) of nonuniformly-spaced data samples has been extended to two dimensions for application to MRI image reconstruction. The 2D nonuniform or generalized FFT (GFFT) was applied to the reconstruction of simulated MRI data collected on radially oriented sinusoidal excursions in...

Introduction: The brain tumor is an abnormal growth of tissue in the brain, which is one of the most important challenges in neurology. Brain tumors have different types. Some brain tumors are benign and some brain tumors are cancerous and malignant. Glioblastoma Multiforme (GBM) is the most common and deadliest malignant brain tumor in adults. The average survival rate for peo...

In this paper, a real-time multi-layer background subtraction based on Gaussian pyramid is proposed for moving object detection. The proposed method models background on two levels: region analysis in the high-resolution level with averaging background model and pixel analysis in the low-resolution level with hierarchical non-parametric kernel density estimation method. The new method has lower...

The success of kernel-based learning methods is heavily dependent on the choice of a kernel function and proper setting of its parameters. In this paper, we optimize the Gaussian kernel for binary-class problems by using centered kernel polarization criterion. This criterion is an extension of kernel polarization and a simplified style of centered kernel alignment. Compared with formulated kern...

Regularized empirical risk minimization using kernels and their corresponding reproducing kernel Hilbert spaces (RKHSs) plays an important role in machine learning. However, the actually used kernel often depends on one or on a few hyperparameters or the kernel is even data dependent in a much more complicated manner. Examples are Gaussian RBF kernels, kernel learning, and hierarchical Gaussian...

Principal Component Analysis (PCA) is a very well known statistical tool. Kernel PCA is a nonlinear extension to PCA based on the kernel paradigm. In this paper we characterize the projections found by Kernel PCA from a information theoretic perspective. We prove that Kernel PCA provides optimum entropy projections in the input space when the Gaussian kernel is used for the mapping and a sample...

Nonparametric kernel methods for estimation of probability densities and point process intensities have long been of interest to researchers in statistics and machine learning. Frequentist kernel methods are widely used, but provide only a point estimate of the unknown density. Additionally, in frequentist kernel density methods, it can be difficult to select appropriate kernel parameters. The ...