# Questions tagged [kernels]

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### Sum of squared Dirichlet kernels

Consider the sum $$S_n=\sum_{h=1}^{n-1}D_{q_n}^2\Bigl(\frac{2\pi h}n\Bigr) =\sum_{h=1}^{n-1}\frac{\sin^2((2q_n+1)\pi h/n)}{\sin^2(\pi h/n)},$$ where $D_{q_n}$ is the Dirichlet kernel and $q_n$ is a ...
1 vote
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### Estimator for the conditional expectation operator with convergence rate in operator norm

Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively. Let $L^2(\pi)$ ...
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### Gaussian lower heat kernel bounds on non-convex bounded domain

I am looking for a proof the following theorem. Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
121 views

### Orthonormal bases in RKHSs via interpolating sequences

Definitions and setting Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
1 vote
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### In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel

For a fixed distribution $p^{in}$ on the input space $\mathbb{R}^{n_0}$, consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$. On this space, ...
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### Pair of positive harmonic functions with negative inner product in Drury-Arveson space

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel ...
47 views

### Sufficient conditions to ensure that a function $P(x,y) := \langle \pi(x),\pi(y)\rangle$ can be represented as $P(x,y) = \phi(\langle x,y\rangle)$

Let $n$ be a positive integer. Give a nonempty collection $\mathcal S$ of subsets of $[n] := \{1,2,\ldots,n\}$, define an nonlinear dot-product by \begin{eqnarray} \langle x,y\rangle_{\mathcal S} := \...
Let $K:X \times X \to \mathbb R$ be a (positive-definite) kernel and let $H$ be the induced reproducing kernel Hilbert space (RKHS). Fix $(x_1,y_1),\ldots,(x_n,y_n) \in X \times \mathbb R$. For $t \ge ... 3 votes 0 answers 353 views ### Reproducing kernel Hilbert space of Matérn kernels I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv) On the top of ... 0 votes 0 answers 154 views ### Is the kernel$\vert d_X - d_Y \vert^p$conditionally negative definite? Given two finite metric spaces$(X,d_X)$and$(Y,d_Y)$, for$p > 0$, define the kernel ($4$-D tensor)$K$on$(X \times Y)^2$by: $$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(... 0 votes 1 answer 377 views ### Proof: If a reproducing kernel exists for a Hilbert space, then it is unique I really want to prove the statement in the title but I'm struggling with it. Here my current state: Proof via contradiction. Let \mathcal{H} be a RKHS with two reproducing kernels k and \hat{k} ... 3 votes 1 answer 334 views ### Minimum upper bound for sum of the entries of the inverse covariance matrix Let x \in \mathbb{R}^n and k is RBF kernel$$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$and let \mathbf{K} be the following n \times n covariance matrix$$\mathbf{K} = \... 2 votes 0 answers 191 views ### Extending Ky Fan's eigenvalues inequality to kernel operators --Migrating from MSE since it might fit better here-- Base result The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as: $$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \... 0 votes 0 answers 223 views ### A new method for processing music scores? I have developed a method and python script: https://github.com/githubuser1983/algorithmic_python_music which allows the user to input a midi file and then chose a few numbers as parameters, and the ... 5 votes 1 answer 338 views ### Why is this nonlinear transformation of an RKHS also an RKHS? I came across this paper (beginning of page 6) where they stated that if f,h\in \mathcal{H}, where \mathcal{H} is an RKHS, then l_{h,f}=\left|f(x)-h(x)\right|^q where q\geq 1 also belongs to ... 1 vote 0 answers 97 views ### Subspace of RKHS generated by kernel mean embeddings Suppose \mathcal{H_k} is a reproducing kernel Hilbert space (RKHS) with reproducing kernel k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}. I am looking for results characterising the ... 4 votes 0 answers 68 views ### What is the native Hilbert space associated with the kernel \frac{\sum \min{(x_i,y_i)}}{\sum \max{(x_i,y_i)}}? In this answer on MSE it is shown that the function$$ K:(\mathbb{R}^{>0})^n\times (\mathbb{R}^{>0})^n\rightarrow\mathbb{R}\,\quad K(x,y)=\frac{\sum_{i=1}^n\min{(x_i,y_i)}}{\sum_{i=1}^n\max{(x_i,... 2 votes 0 answers 59 views ### Johnson filtration and lower central series Let$G$be a group and consider the lower central series: $$G=\gamma_1 G \geq \gamma_2 G=[G,\gamma_1 G]\geq \gamma_3G=[G,\gamma_2G]\geq\cdots.$$ Let$S_g^1$be a compact oriented genus$g$surface ... 1 vote 1 answer 247 views ### Simple example of Hammerstein integral equation I'm currently reading this paper (and working on a similar one). The main goal is to study the Hammerstein integral equation (in$\mathcal{C}(I,E))$: $$x(t) = \int_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\... 8 votes 0 answers 281 views ### Monotonicity of log determinant of Gaussian kernel matrix Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation} be a squared-exponential (Gaussian) kernel, with \sigma,\vartheta>0. Consider, for a set of N ... 3 votes 0 answers 304 views ### Analytic formula for the eigenvalues of kernel integral operator induced by Laplace kernel K(x,x') = e^{-c\|x-x'\|} on unit-sphere in \mathbb R^d Let d \ge 2 be an integer and let X=\mathcal S_{d-1} the unit-sphere in \mathbb R^d. Let \tau_d be the uniform distribution on X. Define a function K:X \times X \to \mathbb R by K(x,y) := ... 1 vote 0 answers 206 views ### Variance-based localized Rademacher complexity for RKHS unit-ball Let \mathscr X be a compact subset of \mathbb R^d (e.g the unit-sphere). Let K: \mathscr X \times \mathscr X \to \mathbb R be a positive kernel function and let \mathscr H_K be the induced ... 1 vote 0 answers 81 views ### Concentration of random Rayleigh quotients Let K be some large square matrix of height N, and u a column vector of height N. Fixing n, take a random set of n indices (with replacement) uniformly from 1,...,N. Let y be the ... 3 votes 1 answer 276 views ### Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations? When we have a system of of n linear equations represented by$$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 ... 2 votes 0 answers 88 views ### 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,... 1 vote 0 answers 85 views ### 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 ... 2 votes 0 answers 329 views ### \ell_\infty-norm covering number of RKHS ball \{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\} For any \epsilon \in (0,1), let N_\infty(\epsilon, \mathcal{H}, R) denote the \epsilon-covering number of the RKHS norm ball \{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\} with respect to the \... 3 votes 1 answer 398 views ### Positive definite kernels involving the \min function I am interested in the positive kernels of the form k(x,y) = \min\{a(x,y), b(x,y)\} (assuming k(x,y) = k(y,x)). Some examples including \min\{x,y\} and \min\{f(x)g(y), f(y)g(x)\}, but are ... 1 vote 0 answers 117 views ### Metric transforms that preserve \ell^1 embeddability Consider a function f from reals to reals such that f, when applied to pairwise Manhattan distances between n points, always results in a set of Manhattan distances. Work by Schoenberg and ... 0 votes 0 answers 30 views ### Help for literature on entrywise invariant kernels I am looking for literature on entrywise invariant kernels. The specific example I have in mind is K:R^{d}\times R^{d}\to R and locally compact groups acting on vector space R^{d}. More precisely ... 1 vote 0 answers 42 views ### When is inverse geodesic distance positive definite (in a compact manifold)? We work on a closed smooth Riemannian manifold (M,g) and let K:M\times M\to \mathbb R\cup\{+\infty\} be a kernel, which we assume to be integrable and lower semicontinuous. We say it is positive ... 2 votes 1 answer 554 views ### Defining measures through products of Markov kernels I am quite puzzled by the expression given in equation 21 (page 10) in this paper, https://arxiv.org/pdf/1802.09188.pdf Its LHS seems to be a measure \nu_n^N and hence I guess it takes as argument ... 0 votes 0 answers 80 views ### How to introduce not-orthonormal base on Reproducing Kernel Hilbert Spaces? I read some tutorial papers and slide，and find that the bases on Reproducing Kernel Hilbert Spaces always be orthonormal. For examples，you can refer to this link for the content about Reproducing ... 0 votes 1 answer 201 views ### Green function of the triangular kernel? What is the green function of the triangular kernel K:$$ K(x,y)=1-|x-y|$$where$x,y\in R$such that$|x-y|<1$? 4 votes 0 answers 211 views ### Studying finite groups with Euclidean geometry? Since each finite group$G$can be considered as a subgroup of the symmetric group, by Cayley's theorem, we might see the elements of$G$as permutations$\pi$. Consider for each$\pi \in G\$ the set: ... 