# Questions tagged [kernels]

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### Variance-based localized Rademacher complexity for RKHS unit-ball

Let $\mathscr X$ be a compact subset of $\mathbb R^d$ (e.g the unit-sphere). Let $K: \mathscr X \times \mathscr X \to \mathbb R$ be a positive kernel function and let $\mathscr H_K$ be the induced ...
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### Concentration of random Rayleigh quotients

Let $K$ be some large square matrix of height $N$, and $u$ a column vector of height $N$. Fixing $n$, take a random set of $n$ indices (with replacement) uniformly from $1,...,N$. Let $y$ be the ...
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### Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b}$$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal}$, we can solve for each component of this vector by ...
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### Kernels with finite dimensional feature spaces

Suppose $x,y \in \mathbb{R}^n$ for some given fixed n. Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
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### Spectral properties of random principal submatrices

Let $M$ be a matrix. A principal submatrix of $M$ is a square matrix obtained by deleting from $M$ the $i$-th column and the $i$-th row, for some number of indices $i$ (the rows and columns are ...
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### When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?

$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...
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### The abc-conjecture over the positive rationals and Levy-Schoenberg kernels?

I am continuing the "abc-adventure" and have a specific question, which needs some explanation: In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3). Consider the ...
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Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by \begin{equation} (Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta) \end{equation} where $\left|A(\alpha,\beta) \right|\le Ce^{-|\... 0answers 64 views ### basis representation of a special sinc kernel I have a functional map from$(x,y) \in \mathbb{R}^2$to another function$f_{x,y}(z,w) \in \mathbb{C}$. (Variables$z,w $range from$-\infty$to$\infty$.) That is, for any pair$x,y$, I get a ... 1answer 106 views ### Theory of integration of Kernel in çinlar probability and stochastic I'm reading the probabilistic book write by çinlar, but I don't understand the Kernel theory, in details:$ (E,\mathcal{E}),(F,\mathcal{F})$are two measurable space $$K:E \times \mathcal{F} \... 1answer 174 views ### How to compute bounding coefficients for McDiarmid's inequality? I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question. Given a ... 1answer 196 views ### Reproducing Kernel Hilbert Spaces with positive kernels In my research I'm dealing with the following question. Let E set, K:E \times E \to \mathbb R a positive type function, and \mathcal H := \mathcal H(1+K) (in the sense of the Moore theorem). ... 0answers 134 views ### Spectrum of Kernel - Discrete orthogonal polynomials Trying to solve a problem, I encounter a Kernel of the form$$K(m,n)= e^{-\frac{\beta}{4} (m+n+1)} \frac{2^{2+\frac{m+n}{2}}}{\sqrt{m! n!}} \frac{\sqrt{\pi}}{n-m} \left[ \frac{1}{\Gamma(-m/2)\Gamma(... 3answers 420 views ### The multiplier algebra of a Reproducing Kernel Hilbert Space and its commutant In my research in the theory of Reproducing Kernel Hilbert Spaces I was concerned with this topic which came up but I could not find a reference on: If$ \mathbb{H} $is an RKHS and we denote the ... 1answer 240 views ### Ideals in exterior algebras over the field with two elements Suppose we have an exterior algebra over$\mathbb{F_2}$, say$R = \Lambda_{\mathbb{F_2}}V$, where$V$is an$n$-dimensional$\mathbb{F}_2$vectorspace. Let$x_1,\ldots,x_n$be a basis of that ... 1answer 791 views ### Duality argument to get$L^\infty-L^2$inequality In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that $$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$ where the norms are$L^p$norms. He states ... 1answer 471 views ### On proof of the conditionally negative definiteness of a kernel Let the kernel be$f(\mathbf{x},\mathbf{y}) = \arccos(\mathbf{x}^T \mathbf{y})$, where$\mathbf{x}$and$\mathbf{y}$are$\ell_2$normalized vectors of the same dimensionality, and$\arccos(\cdot): [-...
I am looking to solve a Fredholm Integral Equation of 1st kind of the form: $\int_{a}^b h(u-u')g(u')du' = f(u)$ where $h$ is a general function. I know from the Hilbert-Schmidt theorem that all I ...