Questions tagged [reproducing-kernel-hilbert-spaces]

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Inverse push-forward on RKHS

I am considering an infinite dimensional separable RKHS $H$ of functions from $E$ to $\mathbb{R}$, where $E$ is any measurable space. I denote by $\phi:E \rightarrow H$ the canonical feature map of $H$...
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Differentiability of functions in reproducing kernel Hilbert space

Consider $\mathcal H_k$ to be the RKHS of a reproducing kernel $k$. I am interested in the differentiability properties of $\mathcal H_k$ as a space of functions. More precisely, is there a link ...
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Pros and cons of Nadaraya–Watson estimator vs. RKHS method?

Recently I've been reading some materials about nonparametric methods. Two methods related to the word "kernel" rasied my interest-- Nadaraya–Watson estimator and RKHS method. What's the ...
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Reproducing kernel Hilbert space norm as a smoothness functional

Let $K:X \times X \rightarrow \mathbb{R}$ be a Mercer kernel with an associated RKHS $H$ then the norm $|f|_H^2$ can be used as a way to ensure that $f$ is smooth in $H$. If i understand correctly, ...
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Closure of balls in Reproducing Kernel Hilbert Space (RKHS)

Let $X \subset \mathbb{R}^m$ be compact, and $k: X\times X \rightarrow \mathbb{R}$ be a universal kernel function, in the sense that the corresponding RKHS $\mathcal{H}_k$ is dense in $C(X)$ under the ...
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prove If $k_1$ and $k_2$ are positive semidefinite kernels then $min\{k_1, k_2\}$ and $max\{k_1, k_2\}$ are psd too.

I can prove for $R+$ the function $min(x,y)$ is a positive semidefinite kernel. But I'm stuck in proving the following statement. Suppose $k_1(x,y)$ and $k_2(x,y)$ are positive semidefinite kernels ...
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reproducing kernel hilbert space notation

I'm trying to understand reproducing kernel Hilbert spaces (RKHSs) from scientific papers, however I don't find any gentle introduction. However, my main problem, at the moment, seems to be to ...
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Moore-Aronszajn Theorem and Mercer theorem for the kernel trick

I have been reading about the RKHS and the kernel trick in Machine Learning mainly from https://ngilshie.github.io/jekyll/update/2018/02/01/RKHS.html (1) and https://arxiv.org/pdf/2106.08443.pdf (2). ...
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Bochner integral on function spaces

Let $(X,\Sigma,\mu)$ be a measure space and $B$ be a Banach space. A Bochner-measurable function $f: X \rightarrow B$ is Bochner integrable if and only if $$\int_{X}\|f(x)\|_{B} d \mu(x)<\infty$$ ...
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pre-image of kernel Hilbert space

Suppose the Hilbert Space $H$ contains all functions $k_w: x\to k(w,x)$, it's the fact that every $w\in R^n$ would have a corresponding in $k_w\in H$, but I'm wondering that is every function in $H$ ...
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nonlinear features mapping to the RKHS

Edited: We usually use linear features mapping in RKHS. But how to handle a nonlinear features mapping? For example: if we are given a non linear mapping \phi:R^d\to R^{d+d(d-1)/2}: x = \begin{...
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Can RKHS of Gaussian kernels over $\mathbb{R}^d$ have a non-zero element which is zero on a linear subspace $R^k\subset R^d$ where $k>0$?

I have been thinking on this problem for at least a day now, and I thought the answer is a resounding no. It seems I am wrong, but wanted to double-check. I came across this: And from my understanding ...
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As is standard in Kernel Ridge Regression, let $K \in \mathbb{R}^{n \times n}$ be the kernel matrix corresponding to some data, and let $Y \in \mathbb{R}^n$ be the vector of predictions, and $\lambda \... • 135 0 votes 0 answers 20 views finding kernel preserving transformations My instructor started talking about kernels and defined a kernel on a set$X$to be a function$K:X \times X \rightarrow \mathbb{R}$that is symmetric and positive (semi) definite in the sense that ... • 395 1 vote 1 answer 56 views What is the dimensionality of$\phi(x)$in terms of$n$in kernel function? Kernel Function:$K(x,y) = (b + x^Ty)^2$is a quadratic polynomial kernel, where$x, y ∈ R^n $and$b > 0$. Provide a feature mapping$\phi(x)$such that$K(x,y) = \phi(x)^T\phi(y)$. What is the ... • 11 0 votes 0 answers 25 views "Learning with Kernels" Theorem 2.10, Mercer's theorem I am have trouble proving one of the theorems listed in the book Learning with Kernels: In particular, it is assumed that$\mu(X) < \infty$, I know that$T_k$is a compact self-adjoint operator on ... • 86 0 votes 0 answers 34 views Prove the uniqueness of this bounded linear operator I'm reading the paper Charles R. Baker. Joint measures and cross-covariance operators. Transactions of the American Mathematical Society, 186:273–289, 1973. I'm stuck in the last part of the proof of ... • 61 0 votes 0 answers 54 views Inversion Formula Gaussian Convolution I am looking at the following 2004 paper by S. Saitoh, called "Approximate real inversion formulas of the Gaussian convolution": https://www.researchgate.net/publication/... 1 vote 1 answer 31 views Prove that$f$is a linear combination of the functions$k_{y_i}(x) = min\{x,y_i\}$. This is exercise 2.4 of "An Introduction to the Theory of Reproducing Kernel Hilbert Spaces" by Paulsen, and it states: Let$y_0 = 0 < y_1 < \dots < y_n$be given and let$f: [0,\...
For given Hilbert space $H$ consider its corresponding antispace, i.e. the vector space endowed with its negative inner product. $\Pi$ is now called a Pontryagin space if it can be written as a direct ...