# 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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### 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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### 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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### 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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### 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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### 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 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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