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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