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]
239 questions
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What is the relationship between a kernel function and the inner product of the corresponding Reproducing Kernel Hilbert Space?
For a positive definite kernel, there exists a Reproducing Kernel Hilbert Space (RKHS) whose reproducing kernel is the kernel function.
My question is, what characteristic of the kernel function decides the inner product of the RKHS.
For example,…
Isa_R
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Are there non-continuous kernels?
In Mercer's theorem context, the kernel has to be a continuous function. But if you look at the RKHS theory, the kernel does not have to be continuous.
Is it possible to find a symmetric, positive semidefinite kernel which is non-continuous?
Thanks
htop
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reproducing Kernel Hilbert Space (RKHS) reproducing property
I'm a bit confused about the reproducing property of an RKHS, especially how a function $f$ say, is represented in the space. Suppose $\cal{X}$ is a set and $\cal{H}$ is a Hilbert space. Given a function $K: \cal{X} \times \cal{X} \rightarrow…
wilf
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Existence of vector $w\in\mathbb{C}^n$ so that $Qw=(f(x_1),\dots,f(x_n))$ for RKHS
Suppose I have a RKHS $\mathcal{H}$ on a set $X$ with kernel $K$. Moreover, othere are $\{x_1,\dots,x_n\}\subset X$ distinct points. We denote by $Q=(K(x_i,x_j))\in\mathbb{R}^{n\times n}$. In An Introduction to the Theory of Reproducing Kernel…
swissy
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RKHS Analog for Non-symmetric Kernel
I'm looking for any theoretical work/paper/development where the notion of Reproducing kernel Hilbert space (RKHS), defined via positive definite symmetric kernel $k(x,y) = k(y,x)$, is extended towards non-symmetrical kernels.
Basically, I'm looking…
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How to describe the Reproducing Kernel Hilbert Space (RKHS), given a kernel?
So, I have the following exercise I need to solve:
"For $x, y \in \mathbb{R}, \: K_1(x,y) = (xy + 1)^2 \:$ and $\: K_2(x,y) = (xy - 1)^2$. Describe the RKHS of $K_1$, $K_2$ and $K_1 + K_2$."
By "describe", I'm pretty sure it's asking for the general…
Berne
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Comparing prediction error on different RKHSs
Let $\Omega \subset \mathbb{R}^d$ and $k_1(\cdot, \cdot), k_2(\cdot, \cdot)$ be two positive definite kernels defined in $\Omega \times \Omega$. Let $\mathcal{H}_1$ and $\mathcal{H}_2$ denote the Reproducing Kernel Hilbert Spaces (RKHS) associated…
sudeep5221
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The kernel function corresponding to a quadratic form rather than inner product
We all know that for a feature map $\Phi$, there exists a kernel function $K_1$ satisfying $\langle\Phi (x),\Phi (y)\rangle=K_1(x,y)$.
For a positive-definite matrix $A$, the quadratic form $\langle\Phi (x),A\Phi (y)\rangle=\langle A^{1/2}\Phi…
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Feature space and Euclidean space
I'm a bit confused with the canonical feature map $\phi_{\mathrm{can}}(x)=k(x, \cdot)$ and explicit feature map $\phi_{\exp }(x)$.
Take the polynomial kernel $k\left(x, x^{\prime}\right)=\left(1+\left\langle x, x^{\prime}\right\rangle\right)^{2}$ as…
zzgsam
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Does RKHS norm preserve inequality in L2?
Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space with reproducing kernel $K$. Assume that $f,g$ are two elements in $L_{2}$ and also in $\mathcal{H}$.
My question is what kind of condition will guarantee that
$$
\|f\|_{\mathcal{H}} \leq…
aprita
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Is a kernel just a symmetric, positive semi-definite, and continuous matrix?
From page 9 of these course notes,
A function $k : \mathcal{X} \times \mathcal{X} \mapsto \mathbb{R}$ is a kernel if
$k$ is symmetric: $k(x,y) = k(y,x)$
$k$ gives rise to a positive semi-definite "Gram matrix," i.e., for any $m \in \mathbb{N}$…
mhdadk
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Is this a valid Kernel?
Please refer the question below, is this a valid kernel? As per my understanding of String kernels, the similarity is in the count of similar strings and not their position, as that can be expressed as a dot product of two feature vectors
Valid…
Yukti Kaura
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Does $\langle h,g\rangle_K<\infty$, where $h \in H(K)$ and $g\in L^2[0,T]$?
It is well known that the reproducing kernel hilber space (RKHS) with reproducing kernel K can be characterized as
$$ H(K) = \left\{f: f \in L^2[0,T], \|f\|_K^2=\langle f,f\rangle_K<\infty\right\} $$
where
$$\langle f,f\rangle_K =…
lz10086lz
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Proof that $K(x,y)=\int\limits_{-\infty}^{\infty}p(t)e^{i(x-y)t}dt$ is a kernel
Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be a piecewise continous, nonnegative function with compath support. Define $K:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{C}$ by
$$K(x,y)=\int\limits_{-\infty}^{\infty}p(t)e^{i(x-y)t}\mathrm{d}t$$.
I…
bayes2021
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kernel centering around mean when samples belong to positive or negative class
suppose I have a kernel matrix of 6x6 .Also i have label information indicating that 1 to 3 samples belong to label(i.e positive label) and 4 to 6 samples do not belong to label(negative label). I want to center the 1-3 samples around the mean of…
