Questions tagged [reproducing-kernel-hilbert-spaces]

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.

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Scale the sequence $a^n$ to converge to $a$

Can i find or assume the existence of a sequence $f_n$ that converges the divergent sequence $a^n$ or the sequence $e^{a\cdot n} $ when they are multiplied? I can write this problem as: Find $f_n$ ...
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Kernel mean embeddings in Reproducing Kernel Hilbert Spaces: linear kernel

I'm reading on kernel mean embeddings, and I got stuck a small detail, but I cannot figure it out, so I'm asking it here :) Some context, we are given a Reproducing Kernel Hilbert Space on a compact ...
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From convergence of orthogonal projection to orthogonal series expansion in reproducing kernel Hilbert spaces.

Introduction: Let $\mathcal{H}$ be a Hilbert space of functions $\Omega\to\mathbb{R}$ with reproducing Kernel $K:\Omega\times\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$, where $K$ is ...
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Embedding for Hilbertian Metric?

Let $X, Y$ be random variables with densites $X = f_x dx$ and $Y = f_ydx$ with respect to the Lesbegue measure. I'm interested in the metric $$d(X, Y)^2 = \frac{1}{2}\int \frac{(f_x - f_y)^2}{f_x+f_y}...
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On the Moore-Aronszajn theorem for reproducing kernel Hilbert spaces

I have a question on the Moore-Aronszajn theorem regarding reproducing kernel Hilbert spaces. The theorem states that a reproducing kernel $K$ on a set $\Omega$ induces a RKHS. A kernel is defined as ...
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Is this map continuous w.r.t. the weak topology?

I am currently working with kernels and I've stumbled upon a map that I really want it to be continuous, but I'm not seeing how to prove it (or disprove it). Consider the following map $\Phi: l_2(\...
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Monotone approximating sequence in RKHS

Consider a kernel $k: X \times X \to \mathbb{R}$ on a topological space $X$, and let $H_k$ denote the associated Reproducing Kernel Hilbert Space (RKHS). It is well-established that the span of ...
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Approximation of a new kernel by a linear combination of previous kernels

From the reference by Knutsen, page 25, Kernel linear independence test is explained Knutsen, Sverre. "Gaussian processes for online system identification and control of a quadrotor." (2019)...
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Positive semidefiniteness of a RKHS

I do not get the derivation on Wikipedia which states that a RHKS is PSD. Specifically, I can see that if the kernel is PSD, then $\langle K_x, K_y \rangle$ must be PSD since $K(x, y)$ is PSD. I do ...
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Inverse of kernel integral operator of Gaussian squared exponential kernel

I am doing research related Gaussian processes and Gaussian process regression. What I would like to know is the inverse of the integral operator of the squared exponential kernel in one dimension, $K ...
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Integrating a product of Gaussian kernels

If we consider the Gaussian kernel $k(x, y) = \exp(-\frac{\Vert x - y \Vert^2}{2\sigma^2})$, then is it true that... $$ \int k(x_i, x)k(x_j, x)dx = k(x_i, x_j) $$ It seems I have seen this before, but ...
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A counterexample to the compactness of inclusion embedding from RKHS to $C(\mathcal{X})$

(I am relatively new to the field of Reproducing Kernel Hilbert Spaces (RKHS) and functional analysis, and I have come across a conceptual discrepancy during my exploration of these subjects.) ...
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Under what kernels and/or conditions does $k(x, x) = k(x, X) k(X, X)^{-1} k(X, x)$?

This question is motivated by a question I'm facing in vector-valued kernel methods (also known as Gaussian Processes and co-krieging). Suppose I have $N$ data $X := \{x_n\}_{n=1}^N$ , where each $x_n ...
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Conditions for transformations of kernels to be kernels

We are given the results that if $(k_n)_{n\in\mathbb{N}}$ are kernels (positive definite symmetric functions) then $k_1+k_2$, $k_1 k_2$ and $\lim_{n\to\infty}k_n$ (if it exists) are kernels, then we ...
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How is this stability bound for the unique interpolant possible? $|s^*(x)|^2\le K(x,x)\|f\|_K\text{cond}_2(G_S)$

If one wants to reconstruct a function $f$ which, we assume is an element of a Hilbertspace $(\mathcal{H}(\Omega,K),(\cdot,\cdot)_K)$ of functions $\Omega\to\mathbb{R}$ with a reproducing Kernel (a.k....
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Mercer's Theorem and Orthonormal Eigenfunctions

Mercer's theorem states that a positive semi-definite kernel $k$ can be decomposed as $k(x,y) = \sum_{m=1}^M \lambda_m \phi_m(x) \phi_m(y)$, where $\lambda_m$ are the eigenvalues and $\phi_m$ are the ...
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How was this bound of the Norm of this Vektor derived? $\|U(x)\|_2^2\le K(x,x)\rho(G_U)$

I was currently reading this article by Robert Schaback and Maryam Pazouki about bases for kernel-based spaces. To ask this question, I'll give a humble Introduction into the tools I'll use. Let $K:\...
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Representer theorem of L2 regularized logistic regression

Let $\left\{\left(x_i, y_i\right)\right\}_{i=1}^n$ be a set of training data, where $x_i \in \mathbb{R}^d$ for all $i$, and $y_i \in\{-1,1\}$. Consider the $l_2$ regularized logistic regression model ...
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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 ...
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How to prove the standard scalar product is a valid kernel?

Let $\mathcal{X}$ be any space. A symmetric function $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ is called a kernel function if for all $n \geq 1, x_1, x_2,..., x_n \in \mathcal{X}$ and $c_1,......
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RKHS norm and $L_\infty$ norm under Gaussian kernel

Given a gaussian kernel $k(x) = \exp(-||x||^2/2)$, let $H_k$ be the associated Reproducing Kernel Hilbert Space(RKHS). My question would be: is it possible to obtain an upper bound of $||f||_{H_k}$ in ...
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Proving that convergence in RKHS implies pointwise convergence without using reproducing property

Let $(\mathcal{H}, \mathcal{K})$ be a reproducing kernel Hilbert space and denote $\mathcal{K}_x := \mathcal{K}(x, \cdot)$. Is there a simple way to prove $f_n \to_\mathcal{H} f$ (shorthand for $\|f_n ...
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How to explicitly write $k(x, )$ in RKHS when using kernel trick?

I am wondering how to explicitly write $K_x(\cdot)$ in RKHS when using kernel trick. ($K_x(\cdot) = K(x, \cdot)$) This is the Moore–Aronszajn theorem. Suppose K is a symmetric, positive definite ...
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right inverse of a linear, bounded, nonnegative, self-adjoint and trace-class operator on closed subspace of a separable Hilbert space

This question is related to Corollary 3 of the paper: Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces by Kenji Fukumizu, et al. Basicly they first defined the ...
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Constructing an RKHS from a Kernel

I'm reading the book "High Dimensional Statistics" by Martin Wainwright just for fun (also as preparation of my PhD in computer science/Machine Learning). In particular, I'm currently ...
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If all features of a kernel $k$ are contained in a RKHS $H$, does then $H(k)\subset H$?

Let $k_1$ and $k_2$ be symmetric positive definite kernels on a set $X$, denote the corresponding RKHS's by $H_1$ and $H_2$, respectively, and also denote the canonical feature maps by $\Phi_1 : X\to ...
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Interpolator with minimum energy: does it act linearly?

I'm interested in constructing a suitable interpolation operator that outputs smooth signals with "minimum energy". Let me clarify. Let $s\ge1$. Let $x_{1},\dots,x_{n}$ be a (finite) ...
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Structure of RKHS induced by a Gaussian kernel

I am studying Reproducing Kernel Hilbert Space, in the context of Maximum Mean Discrepancy. The following points summarize what I've understood up to now If $X$ is a set, $H$ Hilbert space, then $k: ...
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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\...
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Frechet derivative of composition [closed]

Let $f:\mathbb{R} \to \mathbb{R}$ be differentiable at each $x\in \mathbb{R}$ and let $\mathcal{H} \subset \{\mathbb{R}^n\to \mathbb{R}\}$ be a reproducing kernel Hilbert space. Is it true that $f_x:\...
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Is RKHS of Laplace kernel on a subset of sphere a Sobolev space?

Consider $\mathcal{X}\subset \mathbb{S}^n$ and Laplace kernel $k(x,y)=\exp(-\|x-y\|)$. Is the RKHS $H(\mathcal{X})$ given by $k(x,y)=\exp(-\|x-y\|), x,y \in \mathbb{S}^n$ equivalent to Sobolev space $...
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KKT Conditions for SVM Problem

I am reading about SVMs and want to confirm that I understand the optimality conditions. Details below: Consider the $n$ points $x_1, x_2, \dots, x_n$, each with $ d$ dimensions, and consider $ n$ ...
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What is rescaling kernel?

When I was reading the Sub-Bergman Hilbert spaces (doi:10.1016/j.jmaa.2005.12.035), written by Saida Sultanic, I noticed that in page 641, he mentioned: Before we continue our analysis of these ...
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RKHS norm of a constant function?

Given a reproducible kernel Hilbert space (RKHS) with a constant function in this RKHS, is there an analytical expression of the RKHS norm of the constant function?
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Inequality involving kernel matrix and quadratic form

Given data points $x_1,x_2,\dots,x_n\in \mathbb{R}^n$ and kernel $k(.,.): \mathbb{R}^n\times\mathbb{R}^n \to [0,1]$ which satisfies Mercer's theorem, construct the kernel matrix $A$ as $A_{ij} = k(x_i,...
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Mercers condition and an RBF PSD-kernel.

I was given a question in an exam, and it made me realize i don't understand Mercer's condition quite well. I'd be happy for some insight about why my intuition is not right :) for the question we ...
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What does nonlinear mapping $\Phi(x)$ mean? Is it a vector or matrix?

I'm reading a paper about image classification. According to the paper, it says For a given nonlinear mapping $\Phi (x)$, the input data space $\mathbb{R}^{n}$ can be mapped into the feature space $\...
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Weak convergence in a reproducing kernel Hilbert space is equal to pointwise convergence of the sequence

Let $X$ be a set, $k \colon X \times X \to \mathbb R$ be a symmetric positive definite kernel and $H := \overline{\text{span}}(\{ k(x, \cdot): x \in X \})$ the induced real reproducing kernel Hilbert ...
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SVM feature map expansion of sigmoid, Laplace and Student kernels

Denoting the scale and threshold parameters by $\gamma>0$ and $\theta$, in the available literature on SVM, the mapping of a kernel function of two vectors $x$ and $y$ into the reproducing Hilbert ...
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When are RKHS closed under absolute value?

Let $\mathcal{H}\subset \mathbb{R}^X$ a real reproducing kernel Hilbert space (RKHS) with reproducing kernel $k : X\times X\rightarrow\mathbb{R}$. Is then $\mathcal{H}$ closed under absolute value, ...
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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 %\...
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The distance formula used in Kernel K-means in tslearn

I am reading the document of the class tslearn.clustering.KernelKMeans and find its source code. I have questions on the function _compute_dist from the source code which I quote as follows ...
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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 ...
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Centered versus non-centered feature maps in Kernel Ridge Regression

For kernel PCA, I see that people usually center feature maps - hence center the kernel matrix. But for kernel ridge regression, it seems that we do not need to center the feature maps/ kernel matrix ...
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Relationship between kernel method and MMD kernel

Is there any relationship between the two different kernels, since they are both called 'kernel'? Given a kernel $k:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}$, the MMD distance between two ...
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Is the inner product in the RKHS corresponding to the probability space $X$ induced by the Gaussian kernel related to the function integral?

Let $(X, \mathcal{B}, \mu)$ be a probability space, and let $k(\cdot, \cdot): X \times X \rightarrow \mathbb{R}$ be the Gaussian kernel. Denote $\mathcal{H}$ as the RKHS induced by $k$ on $X$. Let $f, ...
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Understanding the definition of Hilbert function space

I was reading the Hilbert function spaces from the chapter 2 of Pick Interpolation and Hilbert Function Spaces by Jim Agler & John E. McCarthy. It says the following- So by Hilbert function space ...
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Meaning of product-like construction in Hilbert space

In [1, Assumption 2] the expression $$|\nabla^3 \ell_z(\theta)[k,h,h]|\leq \sup_{g\in\mathcal{H}}|k\cdot g| \nabla^2\ell_z(\theta)[h,h] $$ is used, where $\ell_z:\mathcal{H}\to \mathbb{R}$ with a ...
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Determinant of random kernel matrix

Let $\kappa(\cdot, \cdot)$ be a positive definite kernel and $X_1, X_2, \dots, X_n$ be $n$ points chosen i.i.d. from a uniform distribution over $[0,1]^d$. Let $K$ denote the random matrix where $K_{...
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Sherman-Morrison formula to update the the inverse of a growing matrix in a iterative way

Hello everyone in the context of my thesis I need to calculate a certain criterion with points arriving sequentially at each time $x_t$. In this criterion, i need to calculate this term at each time $...
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