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Models like support vector machines or Gaussian process regression often require positive semi-definite kernels. These kernels may be based on distance functions. While definiteness is proven for common distances and kernels, a proof for a…

This paper presents new methodology for computationally efficient kernel density estimation. It is shown that a large class of kernels allows for exact evaluation of the density estimates using simple recursions. The same methodology can be…

We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the…

The "point mass singularity" inherent in Newton's law for gravitation represents a major difficulty in accurately determining the potential and forces inside continuous bodies. Here we report a simple and efficient analytical method to…

Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable…

For two decades, reproducing kernels and their associated discrepancies have facilitated elegant theoretical analyses in the setting of quasi Monte Carlo. These same tools are now receiving interest in statistics and related fields, as…

Neural tangent kernels (NTKs) have been proposed to study the behavior of trained neural networks from the perspective of Gaussian processes. An important result in this body of work is the theorem of equivalence between a trained neural…

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of…

As recent literature has demonstrated how classifiers often carry unintended biases toward some subgroups, deploying machine learned models to users demands careful consideration of the social consequences. How should we address this…

We study the problem of distribution to real-value regression, where one aims to regress a mapping $f$ that takes in a distribution input covariate $P\in \mathcal{I}$ (for a non-parametric family of distributions $\mathcal{I}$) and outputs…

Kernel-based learning algorithms are widely used in machine learning for problems that make use of the similarity between object pairs. Such algorithms first embed all data points into an alternative space, where the inner product between…

We present a relativistic generalization of the Wigner inequality for the scalar and pseudoscalar particles decaying to two particles with spin (fermions and photons.) We consider Wigner's inequality with the full spin anticorrelation (with…

An Ensemble Kalman Filter (EnKF, the predictor) is used make a large change in the state, followed by a Particle Filer (PF, the corrector) which assigns importance weights to describe non-Gaussian distribution. The weights are obtained by…

Inverse problems and, in particular, inferring unknown or latent parameters from data are ubiquitous in engineering simulations. A predominant viewpoint in identifying unknown parameters is Bayesian inference where both prior information…

We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries…

This paper provides a new similarity detection algorithm. Given an input set of multi-dimensional data points, where each data point is assumed to be multi-dimensional, and an additional reference data point for similarity finding, the…

Kernel density estimation is a convenient way to estimate the probability density of a distribution given the sample of data points. However, it has certain drawbacks: proper description of the density using narrow kernels needs large data…

We derive a new discrepancy statistic for measuring differences between two probability distributions based on combining Stein's identity with the reproducing kernel Hilbert space theory. We apply our result to test how well a probabilistic…

We construct a family of estimators for a regression function based on a sample following a qdistribution. Our approach is nonparametric, using kernel methods built from operations that leverage the properties of q-calculus. Furthermore,…

Causal effects are usually studied in terms of the means of counterfactual distributions, which may be insufficient in many scenarios. Given a class of densities known up to normalizing constants, we propose to model counterfactual…