Questions tagged [kernels]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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 "...