# Questions tagged [kernels]

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70
questions

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26 views

### Analytic formula for minimum possible error for functions in RKHS ball, on a simple classificaiton problem

Let $X=\mathcal S_{d-1}$ be the unit-sphere in $\mathbb R^d$, and let $K:X \times X \to \mathbb R^d$ be a Mercer kernel (e.g the Laplace kernel). Let $\mathcal H_K$ be the induced RKHS, and for $R \ge ...

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92 views

### Analytic formula for the eigenvalues of kernel integral operator induced by Laplace kernel $K(x,x') = e^{-c\|x-x'\|}$ on unit-sphere in $\mathbb R^d$

Let $d \ge 2$ be an integer and let $X=\mathcal S_{d-1}$ the unit-sphere in $\mathbb R^d$. Let $\tau_d$ be the uniform distribution on $X$. Define a function $K:X \times X \to \mathbb R$ by $K(x,y) := ...

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144 views

### Variance-based localized Rademacher complexity for RKHS unit-ball

Let $\mathscr X$ be a compact subset of $\mathbb R^d$ (e.g the unit-sphere). Let $K: \mathscr X \times \mathscr X \to \mathbb R$ be a positive kernel function and let $\mathscr H_K$ be the induced ...

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27 views

### Concentration of random Rayleigh quotients

Let $K$ be some large square matrix of height $N$, and $u$ a column vector of height $N$. Fixing $n$, take a random set of $n$ indices (with replacement) uniformly from $1,...,N$. Let $y$ be the ...

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**1**answer

111 views

### Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...

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50 views

### Kernels with finite dimensional feature spaces

Suppose $x,y \in \mathbb{R}^n$ for some given fixed n.
Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...

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24 views

### Spectral properties of random principal submatrices

Let $M$ be a matrix. A principal submatrix of $M$ is a square matrix obtained by deleting from $M$ the $i$-th column and the $i$-th row, for some number of indices $i$ (the rows and columns are ...

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47 views

### When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?

$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...

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39 views

### Where can I find general information about the RKHS of a Gaussian kernel?

Let $k$ be the Gaussian, or squared-exponential kernel on $\mathbb R^n$:
$$k(x, y)=e^{-c\lVert x-y\rVert_2^2}$$
where $c$ is some positive constant. This kernel has a reproducing kernel Hilbert space $...

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33 views

### Reproducing kernel Hilbert space of matrix montone function

It is well known that a function $f:(a,b) \to \mathbb{R}$ is matrix-monotone (i.e. $f(A) \leq f(B)$ for any hermitian $A,B \in \mathbb{C}^{n \times n}$ with $A \leq B$ and $\sigma(A),\sigma(B) \...

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106 views

### $\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$

For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...

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**1**answer

92 views

### Positive definite kernels involving the $\min$ function

I am interested in the positive kernels of the form $k(x,y) = \min\{a(x,y), b(x,y)\}$ (assuming $k(x,y) = k(y,x)$). Some examples including $\min\{x,y\}$ and $\min\{f(x)g(y), f(y)g(x)\}$, but are ...

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111 views

### Metric transforms that preserve $\ell^1$ embeddability

Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.
Work by Schoenberg and ...

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24 views

### Help for literature on entrywise invariant kernels

I am looking for literature on entrywise invariant kernels.
The specific example I have in mind is $K:R^{d}\times R^{d}\to R$ and locally compact groups acting on vector space $R^{d}$.
More precisely ...

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26 views

### (random fields / gaussian process): On rewritting a certain expectation as a kernel function

Let $v = (v_1,\ldots,v_n)$ and $(w_{1,1},\ldots,w_{1,n},\ldots, w_{n,m})$ be random vectors with iid coordinates, and also $v$ is independent of $w$, with $w_{i,j} \sim N(0,1/m)$ and $v_j \sim N(0,1/n)...

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49 views

### Bergman Kernel for $D_R (R>0)$

Let $R>0,$ $D_R=\{z\in \mathbb{C} / |z|<R\}$;
I am trying to show that if $f$ holomorphic on $D_R$ and continue on $\bar D_R$, and $w$ is an arbitrary point in $D_R$, then $$f(w)=\frac{R^2}{\pi}\...

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30 views

### When is inverse geodesic distance positive definite (in a compact manifold)?

We work on a closed smooth Riemannian manifold $(M,g)$ and let $K:M\times M\to \mathbb R\cup\{+\infty\}$ be a kernel, which we assume to be integrable and lower semicontinuous. We say it is positive ...

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vote

**1**answer

60 views

### Defining measures through products of Markov kernels

I am quite puzzled by the expression given in equation 21 (page 10) in this paper,
https://arxiv.org/pdf/1802.09188.pdf
Its LHS seems to be a measure $\nu_n^N$ and hence I guess it takes as argument ...

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44 views

### How to introduce not-orthonormal base on Reproducing Kernel Hilbert Spaces?

I read some tutorial papers and slide，and find that the bases on Reproducing Kernel Hilbert Spaces always be orthonormal. For examples，you can refer to this link for the content about Reproducing ...

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**1**answer

180 views

### Green function of the triangular kernel?

What is the green function of the triangular kernel $K$:
$$
K(x,y)=1-|x-y|
$$
where $x,y\in R$ such that $|x-y|<1$?

**4**

votes

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193 views

### Studying finite groups with Euclidean geometry?

Since each finite group $G$ can be considered as a subgroup of the symmetric group, by Cayley's theorem, we might see the elements of $G$ as permutations $\pi$.
Consider for each $\pi \in G$ the set:
...

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91 views

### Irreducible representations and Jaccard Kernel for Groups?

Since each group $G$ can be considered as a subgroup of the symmetric group, we might see the elements of $G$ as permutations $\pi$.
Consider for each $\pi \in G$ the set:
$$X(\pi) := \{ (i,\pi(i)) |...

**5**

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308 views

### Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant 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(a+...

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vote

**1**answer

162 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:\...

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273 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 ...

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100 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

133 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 ...

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**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}\...

**15**

votes

**1**answer

734 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 $|...

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90 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 ...

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353 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\...

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**1**answer

71 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 ...

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**1**answer

106 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 ...

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193 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 ...

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119 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

92 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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**1**answer

986 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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**1**answer

280 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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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 $...

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votes

**1**answer

301 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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64 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

106 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

174 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

196 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**

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**0**answers

134 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

420 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

240 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

791 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
...

**6**

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**1**answer

471 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**

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**0**answers

449 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 ...