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Questions tagged [reproducing-kernel-hilbert-spaces]

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Re-expressing cost function with a kernel

I need to re-express a cost function I will eventually optimize in terms of the provided exponential kernel. The cost function $q(u)$, where $u \in [0, U]$, belongs to the reproducing kernel hilbert ...
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Gradient of square norm in RKHS

Consider a RKHS $\mathcal H$, with continuous reproducing kernel $K$. I am confused regarding the gradient of $\tfrac 12 \Vert \cdot \Vert_{\mathcal{H}} ^ 2$. On the one hand, I'd expect it to be ...
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Find a bound of the biggest radius of hypersphere in convex hull centered on centroid

Setting: We have a probability distribution on a space $\mathcal{X} \subset \mathbb{R}^d$, called $\rho(x)$, and we are given a sample of iid points $S^n = \{x_i\}_{i=1}^n$ from $\rho$. Let $K: \...
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35 views

Kernels and finite maps

Let $\bf{x}, \bf{z} \in \mathbb{R}^n$. If the Gaussian kernel is defined by: $$K(\bf{z}, \bf{x}) = \exp\left( - \frac{\|\bf{z} - \bf{x}\|_2^2}{\sigma^2}\right) $$ I'd like to know if there is a ...
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RKHS for matrix valued input.

The question is related to the link : https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space As I understand the Mercer theorem we can get RKHS. This lead to the kernel trick as mentioned in ...
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SVM: Are kernel functions closed under positive real powers?

If k is a valid kernel (meeting Mercer's conditions) is $k^{\alpha}$ for $\alpha \in \mathbb{R}^{+}$ a valid kernel in general? Any references showing a proof or otherwise would be much appreciated?
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29 views

Kernel function with a feature space equipped with an inner product that is not the dot product

Premise: A function $K: \mathbb R^d \times \mathbb R^d \to \mathbb R$ is called a kernel function on $\mathbb{R}^d$ if there exists a Hilbert space $\mathcal{H}$ and a map $\phi: \mathbb R^d \...
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what does it mean? "the function class becomes an approximation to the Reproducing Kernel Hilbert Space corresponding to the Gaussian kernel.

It seems like embedding. But I am not sure what approximation to RKHS means. Or how to prove something is approximation to RKHS?
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Reducing Kernel Hilbert Space: Reproducing property

If the inner product between tho functions is the integral over R of f(x)g(x)dx and it's equal to f(x), how whould g(x) be in order to satisfy this equality?
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Reproducing Kernel Hilbert Spaces

I don't undestand why: The set of continuos functions from a metric space X→R, C(X), forms a vector space over R using the usual definitions of addition and scalar multiplication.
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How to decide the weight of the locally logistic regression?

I have a problem of how to decide the weight of the logistic regression. My model is as below. For the general logistic regression, we have the likelihood of the model: \begin{align} L(\theta) &=...
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Simple (?) Quesiton on Inner Product in Reproducing Kernel Hilbert Space

I'm following the gentle introduction to Reproducing Kernel Hilbert Spaces From Zero to Reproducing Kernel Hilbert Spaces in Twelve Pages or Less by Hal Daum\'e III. I believe the author fully ...
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1answer
62 views

How to prove that the Gram matrix of the Gaussian kernel has full rank?

I'm trying to prove that, given mutually different points $x_1,\dots,x_m$, the Gram matrix $G$ for the Gaussian kernel has $rank(G)=m$. If I can prove that the Gaussian kernel is strictly positive ...
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What is the meaning of stochastic sampling?

I came across this term in the context of Kernel Methods for Supervised Learning. Subsampling is the selection of a subset from the training set. But what is stochastic subsampling, I understand that ...
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RKHS of a Polynomial Kernel with negative roots

Wikipedia, and actually all books and ressources I could find, define a polynomlial kerenel as: $$ K: x,y \mapsto (x^Ty + c)^2, $$ where $c\ge 0$. What happens if $c < 0$? Take the following ...
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Homogeneous Fredholm Integral Equation of Second Kind with Boundary Conditions

I have come across a claim that I am quite certain is true, but would be benefited from seeing a proof sketch. Suppose we know the following equation: $$ f(z)=\int_0^T K(z,x)f(x)\,\mathrm{d}x, $$ ...
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Each Hilbert space of holomorphic functions on $\mathbb C$ is a reproducing kernel Hilbert space?

I would like to know is the following result true ? "Each Hilbert space of holomorphic functions on $\mathbb C$ is a reproducing kernel Hilbert space". Where a Hilbert space of holomorphic ...
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Uniqueness of Reproducing Kernel Hilbert Space

Digging in the definition of Reproducing Kernel Hilbert Spaces (RKHS) I came across the following example taken from pages 49-51 of [1]: Given the kernel $k(x,y) = \langle x,y\rangle^2$, with $x,y\in ...
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A reproducing kernel Hilbert Space. Barry Simon problem 4.

A reproducing kernel Hilbert Space is a Hilbert space , $H$, of functions , $f$ on a set $E$, so that i) For any $f$, there is $x\in E$ with $f(x)\not=0$ ii) For any $x\in E$, there is $f\in H$ so ...
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Does the Gaussian RKHS contain a nonzero constant function?

I am working with the reproducing kernel Hilbert space $\mathcal{H}$ of functions generated by the Gaussian kernel $k(x,y)=\exp{\left(-\frac{1}{2\sigma^2}|x-y|^2\right)}$ over an interval $[-1,1]$. Is ...
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Correspondence between RKHS and Gaussian process realzation

Consider * the space of realizations of Gaussian process (GP) with zero mean and kernel (covariance function) $k(x, x')$ * the space of functions from RKHS generated by the same kernel $k(x, x')$. ...
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Why $\nu>-1$ in weighted Bergman spaces?

Let $\Omega$ be a non-empty connected open subset of $\mathbb{C}$, let $\mu$ denote the Lebesgue measure on $\Omega$ and let $H(\Omega)$ denote the set of holomorphic functions defined on $\Omega$. ...
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How to prove that the inner space has this property?

Let $A=[a_{i,j}]_{n\times n}$ be a real positive semi-definite type $n\times n$ matrix whose columns are $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_n$. Let those columns span $V \subset R^n$. How ...
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How to prove this equality relating to Fourier-transform and Reproducing-Kernel-Hlibert-Space?

How can we prove following equality? (or, is this equality valid?) Let $x$ and $\zeta$ be members of $\mathbb{R}^d$. And let $K$ be kernel function, that is, $K$ is a symmetric positive-definite ...
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Kernel function from Hausdorff distance

Is there a way to build a positive definite kernel on the space of arbitrary point-sets in $\mathbb{R}^d$ using the Hausdorff distance? I tried the obvious option of setting $K(X, Y) = \exp\{-d_H(X, Y)...
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Derivation of Kernel Function

I'm strugging to find information on how to derive a kernel function or the “feature space”. Let’s assume we have a matrix $x$ of dimension $(n,p)$. We have $x_i^j$, the $i$-th observation of the $j$-...
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eigenvalues and their analogy with frequency

I was reading this document. At page 25, paragraph 2, there was mention of eigenvalues (and frequency; degree of freedom) of a kernel matrix. I am attaching the corresponding image for reader's ...
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125 views

Is there a positive-semidefinite convolution kernel, that is continuous at $0$ but discontinuous elsewhere?

A positive-semidefinite, symmetric convolution kernel on the circle $\mathbb{T}^1$ is a function $k:\mathbb{T}^1\to\mathbb{R}$ such that $k(x)=k(-x)$, and $\sum_{i=1}^n\sum_{j=1}^n k(x_i-x_j)c_i c_j\...
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Linear regression with feature representation confusion - is design matrix column space the feature space?

I am trying to visualise the geometry of linear regression with feature representation. I have a regression problem with $n$ data pairs $\mathcal{D}:=\{(\mathbf{x},y)_{i}\}_{i=1}^{n}$, independent ...
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Invertibility of Grammian in Reproducing Kernel Hilbert Space

Let $\mathcal{H}$ be a Hilbert space with a reproducing kernel $K:X \times X \rightarrow \mathbb{R}$. For any finite sequence $x_1,...,x_n$ of distinct points in $X$ we define Gramm matrix as $$ M(x_1,...
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Orthonormal system $\{ e_n(t)\}_{n=1}^{\infty}$ is complete $\Leftrightarrow$ $k(t,t) = \sum_{n=0}^{\infty}{|e_n(t)|^2}, \forall t \in \Omega$

I want to prove that, in a RKHS (Reproducing Kernel Hilbert Space), being k(t,s) its reproducing kernel: Orthonormal system $\{ e_n(t)\}_{n=1}^{\infty}$ is complete $\Leftrightarrow$ $k(t,t) = \sum_{...
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Are Matérn class kernels universal kernels or not?

This is a question that I can't find the solution. I don't know it is a open question or it is a well-known result that can be attained from several lemmas. Here are the definition of Matérn class ...
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A biliographic inquiry into Fredholm's kernel

I would like to get the exact bibliographic references from the paper(s) by Erik Ivar Fredholm in which the definition(s) of kernel and trace first appear. If someone could even copy the extract from ...
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Does the optimal function in kernel method have a sparse representation?

The kernel least square method aims to solve the following problem, $$f^*=\arg \min_{f\in \mathcal{H}} E_{x,y}[(f(x)-y)^2]+\frac{\lambda}{2}\Vert f\Vert_{\mathcal{H}}$$ , where $\mathcal{H}$ is some ...
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Representer theorem and support vector machines

The objective function of support vector machines can be written as $\sum_{i=1}^n (1-Y_i(K_i^T\alpha+\mu))_+ +\lambda \alpha ^T K \alpha, $ where K is a kernel matrix, $K_i$ is its ith column. How to ...
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RKHS rough definition

Wikipedia provides an intuitive explanation of RKHS as a Hilbert space of functions where point evaluation is a continuous linear functional: Roughly speaking...if two functions $f$ and $g$ in the ...
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Is a Reproducing Kernel Hilbert Space just a Hilbert space equipped with an “indexed basis”?

I haven't studied any functional analysis yet. My linear algebra is pretty good, I think. Consider the tuple $(H, I, \phi)$ where $H$ is a Hilbert space, $I$ is an abstract set, and $\phi:I \to H$ is ...
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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 ...
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Prove strictly positive-definite kernel

It can be shown that the Hamming distance defines a positive semi-definite kernel (see e.g., here). However, is the kernel $$ K(\mathbf{x}, \mathbf{y}) = 1 - \frac{1}{n} \sum_{i=1}^n \mathbb{I}\{x_i = ...
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Which theorem tells us that RKHS is dense in L2?

I have heard this phrase quite a lot now, that RKHS is dense in the space of bounded continuous functions ($\mathcal L_2$). For example, this would be true with $$f(x) = \sum_{i=1}^N \alpha_i K(x_i,...
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Minimizing Hilbert-Norm through Gradient Descent in Parsimonious Online Learning with Kernels

I am conducting research on an algorithm called Parsimonious Online Learning with Kernels (POLK), which uses reproducing kernel hilbert spaces as a means for approximating a function. Here is a link ...
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Statistical learning and differentiation in a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and $\Phi\colon\mathbb{R}^d\to\mathcal{H}$ some function with $K$ being the RKHS kernel. Let $B_i>0$ for all $i=1,2,\ldots ,n$ and we are interested in solving ...
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Characterisation of Banach space $B$, given the RKHS of a Gaussian random variable in $B$.

The setting: Given a Banach space valued Gaussian random variable $X$ (throughout I will assume everything to be centered and separable) we can define its reproducing kernel Hilbert space $$H:=\...
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Completion of span in $C(X)$ (RKHS)

In the construction of Reproducing Kernel Hilbert spaces via the Moore–Aronszajn theorem one uses the completion of the linear span of $\{K_x |\ x\in X\}$, where $K_x(y)=K(x,y)$ and $K$ is some ...
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Why is $\mathbb{R}^3$ not the Reproducing Kernel Hilbert Space defined by kernel $k(x,y)=(x_1y_1+x_2y_2)^2$

Let the input space be $\mathcal{X}=\mathbb{R}^2$, and kernel function $k(x,y)=\langle x,y \rangle^2 = (x_1y_1+x_2y_2)^2$. I can write \begin{align} k(x,y) &= x_1^2 y_1^2 + x_2^2 y_2^2 + 2 x_1 ...
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Linear kernel is strictly positive definite, but corresponding RKHS in not dense in C(R)?

There must be something very basic that I am missing. If I understood correctly from wikipedia, positive-definiteness of kernel $K(\cdot,\cdot)$ is sufficient for universality of this kernel. Then, ...
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1answer
297 views

Reproducing kernel vs. Riesz kernel.

I'm attending an Analysis course and we are studying the Hardy space $H^2$ in the unit disk, from where the concept of RKHS (Reproducing Kernel Hilbert Space) came out. Acording to my notes: By the ...
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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 "...