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8
votes
0answers
190 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using >this< formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= ...
7
votes
1answer
424 views

Which sections of $T^*M\odot T^*M$ have reproducing kernel “primitives”?

Given a smooth reproducing kernel $\kappa:M\times M\rightarrow \mathbb{R}$ on a manifold $M$, we can construct a section, $\alpha_{\kappa}$, of the symmetric tensor product $T^*M\odot T^*M$ by taking ...
6
votes
1answer
1k views

Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...
6
votes
0answers
460 views

The log kernel and Bochner Theorem

I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that $$ L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t) $$ for every $x\in [0,1/2]$. On a structural ground, this ...
5
votes
1answer
117 views

On proof of the conditionally negative definiteness of a kernel

Let the kernel be $f(\mathbf{x},\mathbf{y}) = \arccos(\mathbf{x}^T \mathbf{y})$, where $\mathbf{x}$ and $\mathbf{y}$ are $\ell_2$ normalized vectors of the same dimensionality, and $\arccos(\cdot): ...
5
votes
1answer
671 views

Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition

I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in this paper . Among the bibliography I'm using there are "On the mathematical fundamentals of learning" and ...
5
votes
2answers
567 views

Operators from $L^{\infty}$ to $L^{\infty}$

If $T$ defined as $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded from $L^{\infty}$ to $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le ...
4
votes
1answer
146 views

Ideals in exterior algebras over the field with two elements

Suppose we have an exterior algebra over $\mathbb{F_2}$, say $R = \Lambda_{\mathbb{F_2}}V$, where $V$ is an $n$-dimensional $\mathbb{F}_2$ vectorspace. Let $x_1,\ldots,x_n$ be a basis of that ...
4
votes
0answers
73 views

Level sets of linear combinations of Gaussians

I am trying to work out whether level sets of linear combinations of Gaussian functions are unique. For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let ...
3
votes
1answer
389 views

Does anybody know an estimation of L4 norm of fejer kernel ?

Hi, I need an estimation or an exact closed form expression for the following integral $\int_{0}^{2\pi} K_N^4(s) ds $ where $K_N(s)= \frac{1}{N2\pi} (\frac{sin(Ns/2)}{sin(s/2)})^2$, the Fejer ...
3
votes
2answers
115 views

The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant

In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on: If $ \mathbb{H} $ is an RKHS and we denote the ...
3
votes
0answers
80 views

Measurability for disintegration of a kernel

Let $(x, A) \mapsto P(x, A)$ be a probability kernel whose "target" (wikipedia terminology) is a product space $Y \times Z$, and say both $Y$ and $Z$ are compact metric spaces. For every $x$ there is ...
2
votes
1answer
132 views

Duality argument to get $L^\infty-L^2$ inequality

In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that $$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$ where the norms are $L^p$ norms. He states ...
2
votes
1answer
271 views

Karhunen-Loeve expansion for discrete-time process

Is there a Karhunen-Loeve theorem for discrete-time process? For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...
2
votes
1answer
408 views

Integral kernel of form $e^{-<x,y>^2}$

Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator ...
2
votes
0answers
47 views

Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form $$K(n,m)=\frac{2^{m+n} q^{(m+n+1)}}{n-m} \left[\Gamma(\frac{n}{2}+1)\sin(\pi \frac{n}{2})\Gamma(\frac{m-1}{2}+1) \sin(\pi \frac{m-1}{2}) - ...
2
votes
0answers
101 views

Is every covariance operator the covariance of a measure?

Let $X$ be a topological linear space over $\mathbb R$ which is complete and Hausdorff with a dual space that separates points. Let $k : X^* \to X$ be an arbitrary covariance operator. i.e., any ...
2
votes
0answers
144 views

Gaussian upper bounds on Heat kernel

Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below and let $k(t, x, y)$ be the heat kernel of the Laplace-Beltrami operator. It seems to be standard that for any compact ...
2
votes
0answers
161 views

Cameron-Martin like RKHS

Hello, I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product. What is the RKHS ...
1
vote
1answer
551 views

Eigenfunctions and eigenvalues of the product of two exponential kernels

Consider the following exponential kernel: $k(x_1, x_2) = \exp\left(\frac{|x_1 - x_2|}{L}\right)$, which is symmetric and non-negative definite. By virtue of Mercer's theorem, we have $k(x_1, x_2) ...
1
vote
1answer
283 views

Do kernels provide a basis for a RKHS?

Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, ...
1
vote
0answers
72 views

Solving Fredholm Integral Equations of the first kind

I am looking to solve a Fredholm Integral Equation of 1st kind of the form: $\int_{a}^b h(u-u')g(u')du' = f(u)$ where $h$ is a general function. I know from the Hilbert-Schmidt theorem that all I ...
1
vote
0answers
177 views

system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [closed]

What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$? Define $F$ in the domain where a continuous function exists that behaves so for ...
1
vote
0answers
55 views

Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course. I've seen a reasonable amount of literature about ...
0
votes
2answers
380 views

Kernel of AB if $[A,B]=0$ and $AB\neq0$? [closed]

I have found similar results here and mathematics stack exchange but they all imposed specific conditions that don't suit this problem in particular. The problem is as follows. Let A,B be square ...
0
votes
1answer
93 views

Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...
0
votes
1answer
912 views

Kernel width in Kernel density estimation

Hi, I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions. Also, these samples are just in a metric space ...
0
votes
0answers
46 views

On the positive definiteness of RBF kernel with DTW distance

Consider an RBF kernel that is defined as $K(\mathbf{x},\mathbf{y}) = \exp\left(-\dfrac{d^2(\mathbf{x},\mathbf{y})}{2\sigma^2}\right)$, where $d(\mathbf{x},\mathbf{y})$ is usually chosen as the ...
0
votes
0answers
115 views

Rank of Conjugate Closure of a Subset

How does one find the rank of a conjugate closure of a subset? In particular, I am studying this group: Let $K_n$ be the group with $n$ generators $x_1,\cdots,x_n$ satisfying the relation that each ...
0
votes
0answers
169 views

Finite rank free modules over PIDs

I have two finite rank free modules $M,N$ over a principal ideal domain (the ring of formal complex power series to be exact) and a homomorphism between these two modules $\phi:M\rightarrow N$. Under ...