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]
257 questions
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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
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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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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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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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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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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
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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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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}.$$ This is a "real"…
g g
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Example of an infinite dimensional Hilbert space that is not an RKHS
I have been studying Reproducing Kernel Hilbert Spaces (RKHS). The definition I am using is as follows: An RKHS is a Hilbert space $\mathcal{H}$ of real-valued functions on a set $X$ such that for all $x \in X$, the evaluation functional $E_x :…
ttb
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Convergence theorems for Kernel SVM and Kernel Perceptron
Context Some time ago I asked whether SVMs could work on arbitrary Hilbert spaces , my motivation for asking it was due to my discomfort towards the kernelized version of SVM, which, in my mind, requires that we "restart" SVM in a different Hilbert…
Alek Fröhlich
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Reproducing kernel Hilbert space of set functions
Let $\Omega$ be a finite set. Can we construct a reproducing kernel Hilbert space (RKHS) of real-valued functions $2^\Omega \to \mathbb{R}$? If so, how can we construct one and how is the kernel defined? Thank you!
Alex
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Reproducing kernel Hilbert space induced by $k(x, y) = \delta_{x, y}$, where $\delta$ is the Kronecker delta
I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel $$ k \colon X \times X \to \{ 0, 1 \}, \qquad (x, y) \mapsto %\delta_{x, y} := \begin{cases} 1, & \text{if } x =…
ViktorStein
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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
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Statistical guarantee on the effectiveness of a kernel density estimation
I have recently started studying about kernel density estimations. For reference, if one is given if $(x_1,x_2,...,x_n)$ is an i. i. d. sample drawn from an (unknown) distribution with an unknown $f$ density function, then the kernel density…
Atvin
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