# Questions tagged [kernels]

The kernels tag has no usage guidance.

52
questions

**2**

votes

**0**answers

64 views

### Discriminant of elliptic curve (Frey-Hellegouarch) and positive definite kernels, similarities?

Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(...

**0**

votes

**0**answers

20 views

### Upper bound for eigenvalue of symmetric kernel

Let $V \in L^2(D \times D)$ be symmetric kernel defining the compact and nonnegative integral operator
\begin{equation}\mathcal{V}: L^{2}(D) \rightarrow L^{2}(D), \quad(\mathcal{V} u)(x)=\int_{D} V\...

**1**

vote

**1**answer

141 views

### Does kernel regression preserve monotonicity?

Consider the Kernel regression estimator:
$$\hat{y}(x)=\frac{\sum_{i=1}^n{K(x-x_i)y_i}}{\sum_{i=1}^n{K(x-x_i)}},$$
where $x,x_1,\dots,x_n\in\mathbb{R}^d$, $y_1,\dots,y_n\in\mathbb{R}$, where $K:\...

**2**

votes

**1**answer

68 views

### On conditionally negative definiteness of a kernel

We are interested in the kernel $\phi\left( x-y\right) =\frac{1}{2}\log\left( \cosh\left( x-y\right)
\right) -\log\left( \cosh\left( \frac{x-y}{2}\right) \right)$ on $\mathbb{R} \times \mathbb{...

**5**

votes

**1**answer

255 views

### The abc-conjecture over the positive rationals and Levy-Schoenberg kernels?

I am continuing the "abc-adventure" and have a specific question, which needs some explanation:
In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3).
Consider the ...

**8**

votes

**0**answers

90 views

### Positive definite kernels on categories

I'm wondering if there is any work on studying positive definite kernels on (the objects of a) category. By this I mean for a category $\mathcal{C}$, find a function
$$
K: Ob\mathcal{C} \times Ob\...

**2**

votes

**1**answer

101 views

### Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization)

I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions $\phi$ using (1) Laplace transform of a finite, non-negative Borel measure and also ...

**20**

votes

**1**answer

1k views

### The abc-conjecture as an inequality for inner-products?

The abc-conjecture is:
For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have:
$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...

**0**

votes

**0**answers

47 views

### References for recent books, self-contained papers or downloadable notes that compares spectra of kernel integral operators and kernel matrices

Let $\{x_1,...x_n\}:\Omega \to \mathbb{R}^p$ be $n$ number of iid random vectors, with common probability law $P$, we can assume $dP(x)=f(x)dx$ for nice cases.
I'm trying to find some books/lecture ...

**15**

votes

**1**answer

714 views

### Are primes linearly separable?

Let $X_1,\cdots,X_n$ be finite subsets of some set $Z$. Then the symmetric difference metric space:
$$d(X_i,X_j) = \sqrt{ |X_i|+|X_j|-2|X_i\cap X_j|}$$
can be embedded in Euclidean space. The value $|...

**1**

vote

**0**answers

51 views

### Karhunen-Loeve expansion of vector valued random processes

This thesis (https://www.semanticscholar.org/paper/Karhunen-Loeve-expansions-and-their-applications-Wang/f173dfb99ec4cbd08e779923770466cf1ef3f138) introduces multivariate KL expansion using a ...

**0**

votes

**0**answers

35 views

### Uniform convergence rate of combinations of kernels

I know the uniform convergence rate of kernel density estimator $\hat{f}(x_1,x_2)=\frac{1}{nh_1h_2...h_q}\sum_{i=1}^nk(\frac{X_{1i}-x_1}{h_1})k(\frac{X_{2i}-x_2}{h_2})$ under some smoothness ...

**4**

votes

**0**answers

209 views

### Norms of the Dirichlet kernel

I guess that the following estimates are classical. Let $D_N$ be the $1D$ Dirichlet kernel,
$$
D_N(t)=\frac{\sin((N+\frac12)t)}{\sin (t/2)}.
$$
We have for $1<p<\infty$,
\begin{align}
\Vert D_N\...

**2**

votes

**1**answer

66 views

### Different type of measurability of transition kernel

Let $(E,d)$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{P}(E)$ be the space of all probability measures on $(E,\mathcal{E})$. We eqiup this space with the ...

**0**

votes

**1**answer

84 views

### Does the set of positive definite kernels on some set X form a ring?

Given some non-empty set $X$, does the set of positive definite kernels on $K_X$ form a ring with pointwise addition and multiplication. I am convinced it does not as surely if $k \in K_X$ then we ...

**3**

votes

**0**answers

150 views

### 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 ...

**1**

vote

**0**answers

103 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}...

**2**

votes

**1**answer

90 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 \...

**3**

votes

**1**answer

725 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 ...

**0**

votes

**1**answer

233 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 ...

**11**

votes

**2**answers

3k 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 $...

**-3**

votes

**1**answer

223 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^{-|\...

**1**

vote

**0**answers

59 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 ...

**0**

votes

**1**answer

89 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**

votes

**1**answer

150 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 ...

**0**

votes

**1**answer

191 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). ...

**4**

votes

**0**answers

132 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(...

**2**

votes

**3**answers

366 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 ...

**4**

votes

**1**answer

230 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 ...

**3**

votes

**1**answer

659 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
...

**5**

votes

**1**answer

334 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): [-...

**2**

votes

**0**answers

420 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

**0**answers

185 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\...

**8**

votes

**1**answer

473 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 ...

**0**

votes

**1**answer

289 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 ...

**5**

votes

**1**answer

222 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 ...

**6**

votes

**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 ...

**1**

vote

**0**answers

92 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 ...

**2**

votes

**1**answer

563 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

**0**answers

211 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 ...

**4**

votes

**0**answers

149 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{...

**5**

votes

**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^...

**11**

votes

**0**answers

311 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{...

**0**

votes

**0**answers

205 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 ...

**2**

votes

**1**answer

474 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

**1**answer

889 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) =...

**2**

votes

**0**answers

275 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 ...

**3**

votes

**1**answer

523 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 ...

**7**

votes

**0**answers

661 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

**1**answer

894 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 "...