A reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional, which means that if two functions in the RKHS are close in norm, then they are also pointwise close.
Questions tagged [reproducing-kernel-hilbert-spaces]
264 questions
15
votes
3 answers
Proving the following inequality (positive def. matrix)
I'm trying to prove (or disprove) the following:
$$ \sum_{i=1}^{N} \sum_{j=1}^{N} c_i c_j K_{ij} \geq 0$$
where $c \in \mathbb{R}^N$, and $K_{ij}$ is referring to a kernel matrix:
$$K_{ij} = K(x_i,x_j) = \frac{\sum_{k=1}^{N} \min(x_{ik},…
Norhther
- 217
9
votes
1 answer
Help understanding Reproducing Kernel Hilbert spaces?
I am trying to wrap my head around some concepts of Reproducing Kernel Hilbert Spaces (RKHS) without having a formal background in functional analysis. Since I am trying to form an intuition about what this space is and how it does what it does, I…
J.Galt
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Kernel feature and derivative of kernel feature linearly independent?
Suppose we have a strictly positive definite symmetric kernel $k$ on an open set $\Omega\subset\mathbb R$. By "strictly" I mean that all kernel matrices $(k(x_i,x_j))_{i,j}$ with distinct $x_i$ are positive definite (not only semi-definite). Let us…
amsmath
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8
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Reproducing kernel Hilbert spaces from Sobolev spaces with weight/density functions
I would like to understand which of the statements about the Sobolev space $H^1(\mathbb{R})$ remain true if one introduces a density/weight function in the definition.
Details
The Sobolev space $H^1(\mathbb{R})$ are those square integrable functions…
g g
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2 answers
Why is $L^2$ not a RKHS?
Let's take $L^2[0,1]$ as an example. Obviously $L^2[0,1]$ is not a Hilbert space because you can have a function $f(0)=1$ but equal to $0$ everywhere else, so you don't have an unique element with norm $0$, violating the definition of the inner…
123 456
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0 answers
Can Mercer's theorem fail without positivity?
Let $X = [0,1]$ and $K: X \times X \to \mathbb C$ continuous and self-adjoint, meaning that $K(y, x) = \overline {K(x,y)}
$. It defines a compact, even Hilbert-Schimdt, self-adjoint convolution operator $T_K$ on $L^2(X)$. Call the eigenvalues…
Bart Michels
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6
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2 answers
Functional Analysis: Kernel-Based Approximation
First of all, let me give a basic definition and the problem that we want to solve [both taken from Armin Iske's Book "Approximation Theory and Algorithms for Data Analysis"]:
Problem 8.1. On given interpolation points $X = \{x_1, \dots,…
Hermi
- 1,021
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Intersection of reproducing kernel Hilbert spaces
Let $\mathcal{H} (k), \mathcal{H} (K)$ be two RKHS of functions on the same set, X. It is clear that the intersection of these two spaces, with the norm $\| \cdot \| ^2 := \| \cdot \| _k ^2 + \| \cdot \| _K ^2 $ is also a RKHS.
Recently, I ran…
rtwmartin
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6
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1 answer
The RKHS with kernel $\exp\{x y\}$
I would like to understand the reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ of real valued functions defined on $\mathbb{R}$ generated by or associated with the kernel
$$ K(x,y) = \exp\{x y\} \text{ for }x,y\in\mathbb{R}.$$
This is a "real"…
g g
- 2,799
6
votes
1 answer
Example of an infinite dimensional Hilbert space that is not an RKHS
I have been studying Reproducing Kernel Hilbert Spaces (RKHS). The definition I am using is as follows: An RKHS is a Hilbert space $\mathcal{H}$ of real-valued functions on a set $X$ such that for all $x \in X$, the evaluation functional $E_x :…
ttb
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5
votes
1 answer
Convergence theorems for Kernel SVM and Kernel Perceptron
Context
Some time ago I asked whether SVMs could work on arbitrary Hilbert spaces, my motivation for asking it was due to my discomfort towards the kernelized version of SVM, which, in my mind, requires that we "restart" SVM in a different Hilbert…
Alek Fröhlich
- 209
5
votes
1 answer
Reproducing kernel Hilbert space of set functions
Let $\Omega$ be a finite set. Can we construct a reproducing kernel Hilbert space (RKHS) of real-valued functions $2^\Omega \to \mathbb{R}$? If so, how can we construct one and how is the kernel defined?
Thank you!
Alex
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3 answers
Reproducing kernel Hilbert space induced by $k(x, y) = \delta_{x, y}$, where $\delta$ is the Kronecker delta
I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel
$$
k \colon X \times X \to \{ 0, 1 \}, \qquad
(x, y) \mapsto %\delta_{x, y} :=
\begin{cases} 1, & \text{if } x =…
ViktorStein
- 5,024
4
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0 answers
When I can truncate a function space to a subspace, in a way that non-negative functions stay non-negative?
When I can truncate a function space to a subspace, in a way that non-negative functions stay non-negative?
How I got here (a simple concrete example):
I was working with point-process intensity functions $\rho(x)$ on $x\in\mathbb R^n$. I wanted to…
MRule
- 507
4
votes
0 answers
"Entanglement" of continuous functions via Mercer's theorem
In quantum mechanics, the Schmidt decomposition of a vector $\vert \psi\rangle \in \mathbb{C}^n\otimes \mathbb{C}^m$ is
$$ \vert \psi\rangle = \sum_{i=1}^{\min\{n,m\}} \lambda_i \vert e_i\rangle \otimes \vert f_i\rangle$$
where $\{\vert…
Sam Jaques
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