Given a graph G and a parameter k, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset U ⊆ V (G) of size at most k that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomi...

Partial linear regression is a successful method of dealing with regression tasks. This technique integrates a linear polynomial with a kernel model with the goal of being able to handle nonlinear problems. Kernel models are a kind of instance-based learners and as such do not obtain any intelligible model of the data, which is not the case of linear polynomials. By integrating these two approa...

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain 'native' Hilbert space of multivariate analytic functions. If functions from this space are interpola...

The density estimates considered in this paper comprise a base density and an adjustment component consisting of a linear combination of orthogonal polynomials. It is shown that, in the context of density approximation, the coefficients of the linear combination can be determined either from a moment-matching technique or a weighted least-squares approach. A kernel representation of the corresp...

A methodology and algorithms are proposed for constructing the polynomial chaos expansion (PCE) of multimodal random vectors. An algorithm is developed for generating independent realizations of any multimodal multivariate probability measure that is constructed from a set of independent realizations using the Gaussian kernel-density estimation method. The PCE is then performed with respect to ...

In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with Opkq vertices for the Block Graph Deletion problem. This is a first step to investigate polynomi...

Nyström method has been successfully used to improve the computational efficiency of kernel ridge regression (KRR). Recently, theoretical analysis of Nyström KRR, including generalization bound and convergence rate, has been established based on reproducing kernel Hilbert space (RKHS) associated with the symmetric positive semi-definite kernel. However, in real world applications, RKHS is not a...

In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a perfect graph. However, an open question is the complexity status of the computation of a kernel in such a digraph. Our main contribution is to prove new polyn...

Abstract: Positivity of the prior probability of Kullback-Leibler neighborhood around the true density, commonly known as the Kullback-Leibler property, plays a fundamental role in posterior consistency. A popular prior for Bayesian estimation is given by a Dirichlet mixture, where the kernels are chosen depending on the sample space and the class of densities to be estimated. The Kullback-Leib...

Let k[X]:= k[x 0 , x 1 ,. .. , x n ] be a polynomial algebra over a field k of characteristic zero. We offer an algorithm for calculation of kernel of Weitzenbök derivation d(x i) = x i−1 ,. .. , d(x 0) = 0, i = 1. .. n that is based on an analogue of the well known Casimir elements of finite dimensional Lie algebras. By using this algorithm, the kernel is calculated in the case n < 7.