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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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Conditional Density Estimation in RKHS

I would like to model the conditional density of two real-valued random variable and estimate it using the empirical conditional mean embedding. I am not sure which of these two are correct way of ...
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Possible to define an inner product on tempered distributions of compact support?

I am trying to understand why, in the context of reproducing kernel Hilbert spaces, there seems to always be a square-energy restriction on bandlimited functions in the Paley-Wiener space. (I get why ...
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Is the basis function of an RKHS the same as the eigenfunction of the kernel?

I am trying to derivating the reproducing property of a kernel in an RKHS. However, I meet a conflict as follows: Consider a kernel represented as the sum of a series of basis functions: $k(x,y) = \...
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Operators that preserve RKHSness?

Are there any results on operators that preserve the reproducing property? As an example, orthogonal projection preserves this property (and maps the reproducing kernel to the reproducing kernel as a ...
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Characterization of the RKHS

Let $\mathcal{H}$ be a RKHS on $\mathcal{X}$ with reproducing kernel $K$, and let $f: \mathcal X \to \mathbb{R}$ be a function. Why are the two following equivalent ? $f\in\mathcal{H}$ There exists $...
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Largest RKHS norm of 1-Lipschitz functions on bounded domain and range

The objective is to find the largest RKHS norm of 1-Lipschitz functions on bounded domain and range: $$\sup_{f \in \mathcal{F}} \langle f, f \rangle_\mathcal{H}$$ The domain is the p-dimensional ...
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Let H be a RKHS, show that $f(x)\overline{f(y)} \leq k(x,y)$

I have a question regarding a proof of Aronszajn's inclusion theorem in Paulsen's "An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Let $X$ be a set and $\mathcal{H(K)}$ be the ...
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Reproducing Kernels

I was reading these notes about the uniqueness of the R.K.H.S. (reproducing kernel Hilbert space). https://members.cbio.mines-paristech.fr/~jvert/svn/kernelcourse/notes/uniquenessRKHS.pdf I was just ...
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Is $l_1$-norm isometrically embeddible into Hilbert space?

I am reading Schoenberg's article "METRIC SPACES AND POSITIVE DEFINITE FUNCTIONS". There he proves that $K({\mathbf x}, {\mathbf y}) = e^{-\gamma\|{\mathbf x}-{\mathbf y}\|_p^q}$ is a ...
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What does the statement: "$A$ is a square matrix and $\ker A^{n-1} \neq \ker A^n$ where $n≥2$" imply? [closed]

Currently going over Given the square matrix $A$ and that $\ker(A^2) = \ker(A^3)$ does this imply that $\ker(A^3) = \ker(A^4)$? What about the following statement: $A$ is a square matrix and $\ker A^{...
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Inner product in RKHS

I am reading a paper and am confused by an expression about the inner product. It says that "Given a scalar-valued RKHS $\mathcal{H}$ with a positive definite kernel $k(x,x')$, $\cdots$ and $<\...
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Results of invertibility of a matrix involving the Szego kernel

In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$. Given two sets of points $\{z_1,\ldots,z_n\},\,\{w_1,\ldots,w_n\}\...
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writing empirical covariance operator as the multiplication of sampling operator (elaboration on a paper)

I have been reading this paper (page 85) and have difficulty to understand that how the empirical version of the covariance operator is $\hat{C}_{XX} = \frac{1}{n} S_x^\ast S_x$ can be written as ...
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Can quadratic form be related to integration?

Suppose we have a smooth function $f$ which is defined on $[0,1]$. We pick $n$ points $t_1,\dots, t_n\in[0,1]$ and have the vector $(f(t_1),\dots,f(t_n))^T\in R^n$. We also have a strictly positive ...
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Question about a linear algebra detail of Kernel PCA

As is shown in this question kernel pca eigenproblem and many other refernce materials about kernel PCA. They all point out that the solution of $K^2a_j=\lambda_jnKa_j$ and $Ka_j=\lambda_jna_j$ only ...
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(Why) is the norm of a RKHS positive definite?

$ \newcommand{\real}{\mathbb{R}} $ A $\color{red}{\text{(strictly)}}$ positive definite kernel $k: \real^d\times \real^d \to \real$ satisfies for all $x_i \in \real^d$, $a=(a_1,\dots, a_n)\in \real^d$ ...
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Strictly positive definiteness and functions in the RKHS

A (real-valued) kernel $k$ on $X$ is a positive semi-definite symmetric map $X \times X \to \mathbb{R}$. Specifically, for any $n \in \mathbb{N}$, $x_1,...,x_n \in X$ and $\alpha_1,...,\alpha_n \in \...
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Reproducing Kernel Hilbert Spaces and Yoneda Lemma

In the following articles: https://proceedings.mlr.press/v202/yuan23b/yuan23b.pdf(section 7.4), https://arxiv.org/pdf/2207.02917.pdf(Theorem 4), The Reproducing Kernel Hilbert Spaces(RKHS) are ...
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Feature Orthogonality in RKHS

Let us assume we have linear elements $X = \{x_i\}_i^n$ on the d-sphere. Depending on the number of elements, we may find a configuration that is in expectation, orthogonal, subject to the ...
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Completion of some $C[0,1]$ functions with some inner product is a reproducing kernel Hilbert space

The problem Let $X$ be the space of $C^1[0,1]$ functions with the special property $f(0) = 0$. Consider the inner product defined in the following way: $$\langle f, g \rangle = \int_0^1 f'(x) \...
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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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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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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 $...
Happy Superman's user avatar
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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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