Kernel density estimation via diffusion and the complex exponentials approximation problem

局部拟指数增长模型：核微分方程回归与小鼠肿瘤生长数据

Local Quasi-Exponential Growth Models: Kernel Differential Equation Regression and Mouse Tumor Growth Data

研究针对局部多项式回归处理稀疏数据的挑战，采用微分方程约束回归方法用于局部拟指数增长模型，通过纳入一阶微分方程扩展其稀疏设计能力，减少偏差和方差，经小鼠肿瘤数据等验证该方法有效性。

A Note on Exponential Inequalities in Hilbert Spaces for Spatial Processes with Applications to the Functional Kernel Regression Model

In this manuscript we present exponential inequalities for spatial lattice processes which take values in a separable Hilbert space and satisfy certain dependence conditions. We consider two types of dependence: spatial data under $α$-mixing conditions and spatial data which satisfies a weak dependence condition introduced by Dedecker and Prieur [2005]. We demonstrate their usefulness in the functional kernel regression model of Ferraty and Vieu [2004] where we study uniform consistency properties of the estimated regression operator on increasing subsets of the underlying function space.

Can we trust Bayesian uncertainty quantification from Gaussian process priors with squared exponential covariance kernel?

We investigate the frequentist coverage properties of credible sets resulting in from Gaussian process priors with squared exponential covariance kernel. First we show that by selecting the scaling hyper-parameter using the maximum marginal likelihood estimator in the (slightly modified) squared exponential covariance kernel the corresponding credible sets will provide overconfident, misleading uncertainty statements for a large, representative subclass of the functional parameters in context of the Gaussian white noise model. Then we show that by either blowing up the credible sets with a log…

Free Exponential Families as Kernel Families

Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel instead of the exponential kernel. We use this approach to re-derive several known results and to study further similarities with exponential families and reproductive exponential models.

Diffusion Means and Heat Kernel on Manifolds

We introduce diffusion means as location statistics on manifold data spaces. A diffusion mean is defined as the starting point of an isotropic diffusion with a given diffusivity. They can therefore be defined on all spaces on which a Brownian motion can be defined and numerical calculation of sample diffusion means is possible on a variety of spaces using the heat kernel expansion. We present several classes of spaces, for which the heat kernel is known and sample diffusion means can therefore be calculated. As an example, we investigate a classic data set from directional statistics, for whic…

带非指数核的非平稳Hawkes过程的似然推断

Likelihood inference of the non-stationary Hawkes process with non-exponential kernel

针对经典Hawkes过程的常数基线强度与指数激发核假设过于受限的问题，提出近似方法证明非平稳非指数核Hawkes过程的强度过程渐近遍历，以此推导似然函数的遍历极限，为其极大似然推断的大样本性质提供理论支撑。

扩散映射核岭回归

Diffusion Maps Kernel Ridge Regression

本文提出采用扩散映射（DM）核的核岭回归，证明DM核收敛到热核，其RKHS与高斯核等距同构，且数值验证了DM核在带边界流形监督学习中的优势。

One-sided Cauchy-Stieltjes Kernel Families

This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the exponential families theory. We extend the theory to cover generating measures with support that is unbounded on one side. We illustrate the need for such an extension by showing that cubic pseudo-variance functions correspond to free-infinitely divisible laws without the first moment. We also determine the domain of means, advancing the understanding of Cauchy-Stieltjes kernel families also for compactly supported generating measures.

The Exponent Expansion: An Effective Approximation of Transition Probabilities of Diffusion Processes and Pricing Kernels of Financial Derivatives

A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. Several examples are presented, and the application of these results to increase the efficiency of numerical approaches to derivative pricing is discussed.

Kernelized Diffusion maps

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a graph kernel approach, however this local average construction is known to be cursed by the high-dimension d. In this article, we build a different estimator of the Laplacian, via a reproducing kernel Hilbert space method, which adapts naturally to the regularity of the problem. We provide non-asymptotic statistical rates proving that the kernel estimator we bu…

具有正指数谱的多尺度核的端点稳定区间逆不等式

Endpoint-Stable Interval Inverse Inequalities for Multiscale Kernels with Positive Exponential Spectra

本文为具有正指数谱的多尺度核空间建立了端点稳定的区间逆不等式，通过保留对角能量和控制非对角项，得到与尺度数无关的系数界，并证明了尺度指数的锐性，为确定性采样稳定性提供了定量条件。

Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels

We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of $m \times m$ symmetric positive definite matrices, we show how to efficiently compute by evaluating Pfaffians at specific values …