Family Of Kernels Research Articles

Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product

Let F(νj)={Qmjνj,mj∈(m−νj,m+νj)}, j=1,2, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν1 and ν2. Introduce the set of measures F=F(ν1)⊞F(ν2)={Qm1ν1⊞Qm2ν2,m1∈(m−ν1,m+ν1)andm2∈(m−ν2,m+ν2)}. We show that if F remains a CSK family, (i.e., F=F(μ) where μ is a non-degenerate compactly supported measure), then the measures μ, ν1 and ν2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution are also studied.

Some Limiting Measures Related to Free and Boolean Convolutions

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

Studies on the Marchenko–Pastur Law

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

Asymptotic results for recursive multivariate associated-kernel estimators of the probability density mass function of a data stream

. In this article, our central focus is to investigate the non parametric estimator of the probability density or mass function of a data stream by using the general family of kernels, also called multivariate associated kernels. Being able to estimate categorial, count, discrete, and (semi)continuous distributions, these multivariate associated kernel estimators unify several particular cases. Within this framework, we first introduce a recursive estimator for non classical associated kernels. Under reasonable assumptions, we subsequently exhibit their asymptotic results, such as the local uniform L q -consistency, the pointwise asymptotic normality and finally the strong global consistency. Illustrative examples of associated kernels are revisited.

Compressing spectral kernels in Gaussian Process: Enhanced generalization and interpretability

The modeling capabilities of a Gaussian Process (GP), such as generalization, nonlinearity, and smoothness, are largely determined by the choice of its kernel. A popular family of kernels for GPs, the spectral mixture (SM) kernels, have the desirable property that with a large number of spectral components they can approximate any stationary kernel. However, using a large number of SM components increases the risk of overfitting and hinders interpretability. To overcome these challenges, we propose a compression algorithm incorporating component pruning and component merging for GPs. Here SM components with small signal variance are removed, and a moment-matching merge method is proposed to further reduce the number of SM components. The main novelty of the proposed method is a similarity measure between SM components based on their normalized cross-correlation, which is related to the Bhattacharyya coefficient. We derive a greedy GP compression algorithm and perform a comparative evaluation over various learning tasks in terms of forecasting performance and compression capability. Results substantiate the beneficial effect of the method, both in terms of generalization and interpretability.11Source code:https://github.com/ck2019ML/Gaussian-Process-Model-Compression.

On the expressivity of embedding quantum kernels

One of the most natural connections between quantum and classical machine learning has been established in the context of kernel methods. Kernel methods rely on kernels, which are inner products of feature vectors living in large feature spaces. Quantum kernels are typically evaluated by explicitly constructing quantum feature states and then taking their inner product, here called embedding quantum kernels. Since classical kernels are usually evaluated without using the feature vectors explicitly, we wonder how expressive embedding quantum kernels are. In this work, we raise the fundamental question: can all quantum kernels be expressed as the inner product of quantum feature states? Our first result is positive: Invoking computational universality, we find that for any kernel function there always exists a corresponding quantum feature map and an embedding quantum kernel. The more operational reading of the question is concerned with efficient constructions, however. In a second part, we formalize the question of universality of efficient embedding quantum kernels. For shift-invariant kernels, we use the technique of random Fourier features to show that they are universal within the broad class of all kernels which allow a variant of efficient Fourier sampling. We then extend this result to a new class of so-called composition kernels, which we show also contains projected quantum kernels introduced in recent works. After proving the universality of embedding quantum kernels for both shift-invariant and composition kernels, we identify the directions towards new, more exotic, and unexplored quantum kernel families, for which it still remains open whether they correspond to efficient embedding quantum kernels.

On Generalized t-Transformation of Free Convolution

A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t=(a,b))-deformed free convolution, is introduced for a∈R and b>0. In this article, some results of t-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under t-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t-deformed free Gaussian CSK family and members of the t-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t-deformed free convolution.

On the fourth-order hybrid beta polynomial kernels in kernel density estimation

This paper introduces a novel family of fourth-order hybrid beta polynomial kernels tailored for statistical analysis. The efficacy of these kernels is evaluated using two principal performance metrics: asymptotic mean integrated squared error (AMISE) and kernel efficiency. Comprehensive assessments were conducted using both simulated and real-world datasets, enabling a thorough comparison with conventional fourth-order polynomial kernels. The evaluation process entailed computing AMISE and efficiency metrics for both the hybrid and classical kernels. Consistently, the results illustrated the superior performance of the hybrid kernels over their classical counterparts across diverse datasets, underscoring the robustness and effectiveness of the hybrid approach. By leveraging these performance metrics and conducting evaluations on simulated and real world data, this study furnishes compelling evidence supporting the superiority of the proposed hybrid beta polynomial kernels. The heightened performance, evidenced by lower AMISE values and elevated efficiency scores, strongly advocates for the adoption of the proposed kernels in statistical analysis tasks, presenting a marked improvement over traditional kernels.

Cauchy-Stieltjes Kernel families: Some properties

In the setting of the theory of Cauchy-Stieltjes Kernel (CSK) families, we present in this article some properties related to the Marchenko–Pastur and Semicircle laws. Let K+(μ)={Qmμ(dx);m∈(m1μ,m+μ)} be the CSK family generated by a non degenerate probability measure μ with support bounded from above. Firstly, for 0<t≠1 such that Qmμ⊞t is well defined, we prove that if Qmμ⊞t∈K+(μ) then the measure μ is a scale transformation of the Marchenko–Pastur law. Secondly, consider Tβ:x⟼x+β, for β∈R∖{0}. We prove that if Tβ(Qmμ)∈K+(μ), then the measure μ is an affine transformation of the Semicircle law.

Mixing Sinc kernels to improve interpolations in smoothed particle hydrodynamics without pairing instability

ABSTRACT The smoothed particle hydrodynamic technique is strongly based on the proper choice of interpolation functions. This statement is particularly relevant for the study of subsonic fluxes and turbulence, where inherent small errors in the averaging procedures introduce excessive damping on the smallest scales. To mitigate these errors, we can increase both the number of interpolating points and the order of the interpolating kernel function. However, this approach leads to a higher computational burden across all fluid regions. Ideally, the development of a single kernel function capable of effectively accommodating varying numbers of interpolating points in different fluid regions, providing good resolution and minimal errors would be highly desirable. In this work, we revisit and extend the main properties of a family of interpolators called Sinc kernels and compare them with the widely used family of Wendland kernels. We show that a linear combination of low- and high-order Sinc kernels generates good-quality interpolators, which are resistant to pairing instability while maintaining good sampling properties in a wide range of neighbour interpolating points, 60 ≤ nb ≤ 400. We show that a particular case of this linear mix of Sincs produces a well-balanced and robust kernel that improves previous results in the Gresho–Chan vortex experiment even when the number of neighbours is not large, while yielding a good convergence rate. Although such a mixing technique is ideally suited for Sinc kernels owing to their excellent flexibility, it can be easily applied to other interpolating families such as the B-splines and Wendland kernels.

Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators.

We present incomplete gamma kernels, a generalization of Locally Optimal Projection (LOP) operators. In particular, we reveal the relation of the classical localized L1 estimator, used in the LOP operator for point cloud denoising, to the common Mean Shift framework via a novel kernel. Furthermore, we generalize this result to a whole family of kernels that are built upon the incomplete gamma function and each represents a localized Lp estimator. By deriving various properties of the kernel family concerning distributional, Mean Shift induced, and other aspects such as strict positive definiteness, we obtain a deeper understanding of the operator's projection behavior. From these theoretical insights, we illustrate several applications ranging from an improved Weighted LOP (WLOP) density weighting scheme and a more accurate Continuous LOP (CLOP) kernel approximation to the definition of a novel set of robust loss functions. These incomplete gamma losses include the Gaussian and LOP loss as special cases and can be applied to various tasks including normal filtering. Furthermore, we show that the novel kernels can be included as priors into neural networks. We demonstrate the effects of each application in a range of quantitative and qualitative experiments that highlight the benefits induced by our modifications.

New limit theorems related to Fermi convolution

This paper is a continuation of study of Fermi convolution from the perspective of Cauchy-Stieltjes Kernel (CSK) families. By the use of variance function machinery, we prove some new limit theorems related to Fermi convolution. We give an approximation of elements of the CSK family generated by the Fermi Gaussian distribution. We also provide a new limit theorem related to Fermi convolution and involving free multiplicative convolution.

Strictly positive definite non-isotropic kernels on two-point homogeneous manifolds: the asymptotic approach

We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details.

Fermi convolution and variance function

In this paper, we determine the effect of the Fermi convolution power on the variance function of a Cauchy-Stieltjes Kernel (CSK) family. We then use the criteria of convergence for a sequence of variance functions to give an approximation of elements of the CSK family generated by the Fermi Poisson distribution.

TAKDE: Temporal Adaptive Kernel Density Estimator for Real-Time Dynamic Density Estimation.

Real-time density estimation is ubiquitous in many applications, including computer vision and signal processing. Kernel density estimation is arguably one of the most commonly used density estimation techniques, and the use of "sliding window" mechanism adapts kernel density estimators to dynamic processes. In this article, we derive the asymptotic mean integrated squared error (AMISE) upper bound for the "sliding window" kernel density estimator. This upper bound provides a principled guide to devise a novel estimator, which we name the temporal adaptive kernel density estimator (TAKDE). Compared to heuristic approaches for "sliding window" kernel density estimator, TAKDE is theoretically optimal in terms of the worst-case AMISE. We provide numerical experiments using synthetic and real-world datasets, showing that TAKDE outperforms other state-of-the-art dynamic density estimators (including those outside of kernel family). In particular, TAKDE achieves a superior test log-likelihood with a smaller run-time.

Positive Definite Kernels for Partitions

This paper presents a comprehensive exploration of various families of positive definite kernels for comparing partitions. It not only reviews existing examples from the literature but also introduces novel classes of positive definite kernels. These new classes include kernels based on agreement and ones designed using the concept of hidden variables. The study also focuses on assessing the compatibility of these kernels with structural properties that capture the intrinsic notion of proximity between partitions. Notably, agreement-based kernels are demonstrated to align well with this notion. Moreover, the paper provides two generic procedures for designing hidden-feature-based kernels that also adhere to the specified structural properties.

Integral-based non-local approach to ductile damage and mixed-mode fracture

The present study focuses on the application of an integral-based approach for constructing non-local ductile damage models. The incorporation of non-locality into the damage accumulation rule enhances the modeling framework and yields simulation results that are physically reasonable, avoiding the issue of pathological mesh-dependence. To achieve a precise depiction of damage accumulation and fracture under mixed-mode I/II loading, novel delocalization kernels are proposed. These kernels explicitly consider the heterogeneity of stresses and strains in the fracture process zone; the new kernels are categorized into stress-based and strain-based families. Furthermore, rigorous receiver-based normalization procedures and physically meaningful source-based normalizations are explored for the developed kernels. The outcome of this study is the modeling tool, suitable for end-to-end simulations of damage accumulation and fracture. Using a well-calibrated material model, the practical applicability of the new approach is demonstrated by performing finite element analysis on fracture tests conducted on compact tension–shear specimens. In contrast to the conventional delocalization kernels, the newly introduced families of kernels offer an additional fitting parameter, valuable in accurately describing the structural behavior under mixed-mode loading conditions. In particular, the proposed approach enables control over the shape of the predicted KIc–KIIc diagram.

Adaptive Robotic Information Gathering via non-stationary Gaussian processes

Robotic Information Gathering (RIG) is a foundational research topic that answers how a robot (team) collects informative data to efficiently build an accurate model of an unknown target function under robot embodiment constraints. RIG has many applications, including but not limited to autonomous exploration and mapping, 3D reconstruction or inspection, search and rescue, and environmental monitoring. A RIG system relies on a probabilistic model’s prediction uncertainty to identify critical areas for informative data collection. Gaussian processes (GPs) with stationary kernels have been widely adopted for spatial modeling. However, real-world spatial data is typically non-stationary—different locations do not have the same degree of variability. As a result, the prediction uncertainty does not accurately reveal prediction error, limiting the success of RIG algorithms. We propose a family of non-stationary kernels named Attentive Kernel (AK), which is simple and robust and can extend any existing kernel to a non-stationary one. We evaluate the new kernel in elevation mapping tasks, where AK provides better accuracy and uncertainty quantification over the commonly used stationary kernels and the leading non-stationary kernels. The improved uncertainty quantification guides the downstream informative planner to collect more valuable data around the high-error area, further increasing prediction accuracy. A field experiment demonstrates that the proposed method can guide an Autonomous Surface Vehicle (ASV) to prioritize data collection in locations with significant spatial variations, enabling the model to characterize salient environmental features.

Swarming: hydrodynamic alignment with pressure

We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering toward a weighted average heading. We consider the class of so-called p p -alignment hydrodynamics, based on 2 p 2p -Laplacians and weighted by a general family of symmetric communication kernels. The main new aspect here is the long-time emergence behavior for a general class of pressure tensors without a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as “entropic”, and prove the flocking of p p -alignment hydrodynamics, driven by singular kernels with a general class of entropic pressure tensors. These results indicate the rigidity of alignment in driving long-time flocking behavior despite the lack of thermodynamic closure.

The Introductory Analysis on Fourier Series

Numerical analysis benefits from spectral methods. The Fourier series and Chebyshev polynomials were initially used to solve ordinary differential equations by Cornelius Lanczos. For the first time, spectral approaches were employed by Kreiss, Oliger, Orszag, and other scholars in the 1970s to resolve the partial differential equations of fluid mechanics. In Fourier analysis, general functions are expressed as the sums of a variety of fundamental function options, such as sines and cosines.This paper gives an introductory analysis on Fourier Series. Starting with the basic definition of Fourier coefficients and Fourier Series, the paper introduces the concept and properties of convolution. It also talks about different kernels and good kernel family, analyzing the relationship between these kernels and the original function. Then the paper will combine these ideas and prove the uniqueness and convergence of Fourier Series.