# Questions tagged [kernels]

The kernels tag has no usage guidance.

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### Simple (?) question on inner product in reproducing kernel Hilbert space

I'm following the gentle introduction to Reproducing Kernel Hilbert Spaces From Zero to Reproducing Kernel Hilbert Spaces in Twelve Pages or Less by Hal Daumé III. I believe the author fully ...

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67 views

### Preserving the strictly total positivity of special bases by using radial basis functions

Let $\left[\alpha,\beta\right]$ be a non-empty interval and consider an $\left(n+1\right)$-dimensional subspace $\mathbb{S}_{n}$ of $C^n\left(\left[\alpha,\beta\right]\right)$ such that $\mathbb{S}_{n}...

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**1**answer

85 views

### Equivalence of RKHS with high probability

Suppose you have two PSD kernels $k(x,y)$ and $k'(x,y)$. Let $\mathcal{H}_k(\mathcal{D})$ and $\mathcal{H}_{k'}(\mathcal{D})$ be the corresponding RKHS's.
Now, if we know that for all $f \in \...

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26 views

### Are Matérn class kernels universal kernels or not?

This is a question that I can't find the solution. I don't know it is a open question or it is a well-known result that can be attained from several lemmas.
Here are the definition of Matérn class ...

**3**

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**1**answer

295 views

### Earth movers distance (EMD) between two multivariate normals. Is it negative definite distance?

I was looking at the closed form formula for 2-Wassersteins distance for multivariate normal distribution on Wikipedia. https://en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions
It has a ...

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94 views

### concentration inequalities on RKHS?

Let $\mathcal{H}$ be an RKHS and $\phi$ is defined such that $\langle \phi(x), l\rangle_{\mathcal{H}} = l(h(x),h'(x))$, where $h$ and $h'$ are two functions on a function space $\mathcal{F}$. I'm ...

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**1**answer

176 views

### How can Kernel functions make a Grassmann manifold into an Euclidean vector space?

I'm trying to read a paper called "Graph Embedding Discriminant Analysis on Grassmannian Manifolds for Improved Image Set Matching" and I came across a sentence that confused me (the last one):
A ...

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**2**answers

2k views

### Prove that matrix is positive definite

I faced a hard question in kernel methods theory, which I can't answer for about one week. Initially it was formulated in terms of positive valued functions, but it could be reformulated easier:
Let $...

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**1**answer

166 views

### Exponential decay of kernel

Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...

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vote

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55 views

### basis representation of a special sinc kernel

I have a functional map from $(x,y) \in \mathbb{R}^2$ to another function $f_{x,y}(z,w) \in \mathbb{C}$. (Variables $z,w $ range from $-\infty$ to $\infty$.) That is, for any pair $x,y$, I get a ...

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**1**answer

76 views

### Theory of integration of Kernel in çinlar probability and stochastic

I'm reading the probabilistic book write by çinlar, but I don't understand the Kernel theory, in details:
$ (E,\mathcal{E}),(F,\mathcal{F})$ are two measurable space
$$K:E \times \mathcal{F} \...

**2**

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**1**answer

133 views

### How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question.
Given a ...

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**1**answer

178 views

### Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...

**3**

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119 views

### Spectrum of Kernel - Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form
$$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(...

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**3**answers

317 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 ...

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**1**answer

200 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 ...

**2**

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**1**answer

440 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
...

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186 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 ...

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243 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): [-...

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357 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 ...

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184 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 $x\...

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**1**answer

468 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 ...

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**1**answer

222 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 ...

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107 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 ...

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**1**answer

2k 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 ...

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73 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 ...

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**1**answer

441 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 $\{-1,1\}$. ...

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189 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 ...

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137 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 $\mathcal{...

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**2**answers

556 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 $n\...

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**2**answers

1k 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 \|T\|_{L^...

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130 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 $...

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281 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=
\begin{...

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198 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 ...

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**1**answer

408 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, \...

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**1**answer

852 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) =...

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230 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 ...

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**1**answer

493 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 ...

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608 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**

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**1**answer

833 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 "...

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**1**answer

436 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 $T:C(\...

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**1**answer

1k 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 (...