A method of counts-in-cells analysis of galaxy distribution is investigated with arbitrary smoothing functions in obtaining the galaxy counts. We explore the possiblity of optimizing the smoothing function, considering a series of m-weight Epanechnikov kernels. The popular top-hat and Gaussian smoothing functions are two special cases in this series. In this paper, we mainly consider the second...

t (x)2, as was to be shown. B Learning ✏ via Maximum-Likelihood In this section, we provide an overview of how ✏ can be learned from training data in a principled manner; the details can be found in [20, Section 4.3] and [6, Section 5]. Throughout this appendix, we assume that the kernel matrix is parametrized by a set of hyperparameters ✓ (e.g., ✓ = (⌫, l) for the Mátern kernel), and ✏. Let ȳ ...

Let Xn, n ∈ IN, be a stationary sequence of associated random variables with uniform distribution on [0, 1] and F the distribution function of (X1, Xk+1), for fixed k ∈ IN. We introduce a kernel estimator for F and study its asymptotic properties and moments, characterizing their convergence rates. From these we derive the optimal rate for the bandwidth, which is of order n. Conditions are also...

The number of modes (also known as modality) of a kernel density estimator (KDE) draws lots of interests and is important in practice. In this paper, we develop an inference framework on the modality of a KDE under multivariate setting using Gaussian kernel. We applied the modal clustering method proposed by [1] for mode hunting. A test statistic and its asymptotic distribution are derived to a...

This project was done at the Cambridge Machine Learning Group, as part of the larger effort to build an Automated Statistician. Given a data set, the Automated Statistician should run different methods to suggest interpretable hypotheses and the potential models to use for this data. In this project, we have shown that automatic kernel discovery can be achieved for GP classification. We impleme...

We investigate training and using Gaussian kernel SVMs by approximating the kernel with an explicit finitedimensional polynomial feature representation based on the Taylor expansion of the exponential. Although not as efficient as the recently-proposed random Fourier features [Rahimi and Recht, 2007] in terms of the number of features, we show how this polynomial representation can provide a be...

Several learning algorithms for topographic map formation have been introduced that adopt overlapping activa-tion regions, rather than Voronoiregions, usually in the form of kernel functions. We review and introduce a numberof fixed point rules for training homogeneous, heteroscedastic but otherwise radially-symmetric Gaussian kernel-based topographic maps, or kernel topographic...

We describe a formula for the Taylor series expansion of the Gaussian kernel around the origin of Rn × R.

Although the real reproducing kernels are used in an increasing number of machine learning problems, complex kernels have not, yet, been used, in spite of their potential interest in applications such as communications. In this work, we focus our attention on the complex gaussian kernel and its possible application in the complex Kernel LMS algorithm. In order to derive the gradients needed to ...