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Understanding how neural networks transform input data across layers is fundamental to unraveling their learning and generalization capabilities. Although prior work has used insights from kernel methods to study neural networks, a global…

We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complex-valued kernel least-mean-square (gCKLMS). We borrow from the new results on widely linear reproducing kernel Hilbert space…

This paper investigates the critical role of eigenalignments between the kernel matrix and learning targets in achieving robust generalization in learning problems. We establish a direct connection between generalization performance in…

The study of networks has witnessed an explosive growth over the past decades with several ground-breaking methods introduced. A particularly interesting -- and prevalent in several fields of study -- problem is that of inferring a function…

The main purpose is to estimate the regression function of a real random variable with functional explanatory variable by using a recursive nonparametric kernel approach. The mean square error and the almost sure convergence of a family of…

Contemporary wisdom based on empirical studies suggests that standard recurrent neural networks (RNNs) do not perform well on tasks requiring long-term memory. However, precise reasoning for this behavior is still unknown. This paper…

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back…

Non-parametric representations of dynamical systems based on the image of a Hankel matrix of data are extensively used for data-driven control. However, if samples of data are missing, obtaining such representations becomes a difficult…

Kernel embeddings have emerged as a powerful tool for representing probability measures in a variety of statistical inference problems. By mapping probability measures into a reproducing kernel Hilbert space (RKHS), kernel embeddings enable…

An interesting approach to analyzing neural networks that has received renewed attention is to examine the equivalent kernel of the neural network. This is based on the fact that a fully connected feedforward network with one hidden layer,…

Functional regression is very crucial in functional data analysis and a linear relationship between scalar response and functional predictor is often assumed. However, the linear assumption may not hold in practice, which makes the methods…

The Kernel Complex Lemma states that given commutative diagram with exact rows and exact columns which covers the region under a $\Gamma$-shape, then the kernel sequence on the top and the kernel sequence at the left have in each position…

Graph kernels are kernel methods measuring graph similarity and serve as a standard tool for graph classification. However, the use of kernel methods for node classification, which is a related problem to graph representation learning, is…

An extension of reproducing kernel Hilbert space (RKHS) theory provides a new framework for modeling functional regression models with functional responses. The approach only presumes a general nonlinear regression structure as opposed to…

Kernels are powerful and versatile tools in machine learning and statistics. Although the notion of universal kernels and characteristic kernels has been studied, kernel selection still greatly influences the empirical performance. While…

This paper generalizes regularized regression problems in a hyper-reproducing kernel Hilbert space (hyper-RKHS), illustrates its utility for kernel learning and out-of-sample extensions, and proves asymptotic convergence results for the…

Kernel methods, particularly kernel ridge regression (KRR), are time-proven, powerful nonparametric regression techniques known for their rich capacity, analytical simplicity, and computational tractability. The analysis of their predictive…

In this article we shall study the analytic theory and the representation theoretic interpretations of Hankel transforms and fundamental Bessel kernels of an arbitrary rank over an archimedean field.

A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when…

We propose a kernel-based partial permutation test for checking the equality of functional relationship between response and covariates among different groups. The main idea, which is intuitive and easy to implement, is to keep the…