# Questions tagged [kernels]

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### What is lost after RKHS embedding of the L1 space?

We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
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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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### 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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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 $\... 3 votes 1 answer 152 views ### Positive definite kernels involving the$\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)\}$, but are ... 1 vote 0 answers 115 views ### Metric transforms that preserve$\ell^1$embeddability Consider a function$f$from reals to reals such that$f$, when applied to pairwise Manhattan distances between$n$points, always results in a set of Manhattan distances. Work by Schoenberg and ... 0 votes 0 answers 25 views ### Help for literature on entrywise invariant kernels I am looking for literature on entrywise invariant kernels. The specific example I have in mind is$K:R^{d}\times R^{d}\to R$and locally compact groups acting on vector space$R^{d}$. More precisely ... 1 vote 0 answers 32 views ### When is inverse geodesic distance positive definite (in a compact manifold)? We work on a closed smooth Riemannian manifold$(M,g)$and let$K:M\times M\to \mathbb R\cup\{+\infty\}$be a kernel, which we assume to be integrable and lower semicontinuous. We say it is positive ... 1 vote 1 answer 180 views ### Defining measures through products of Markov kernels I am quite puzzled by the expression given in equation 21 (page 10) in this paper, https://arxiv.org/pdf/1802.09188.pdf Its LHS seems to be a measure$\nu_n^N$and hence I guess it takes as argument ... 0 votes 0 answers 52 views ### How to introduce not-orthonormal base on Reproducing Kernel Hilbert Spaces? I read some tutorial papers and slide，and find that the bases on Reproducing Kernel Hilbert Spaces always be orthonormal. For examples，you can refer to this link for the content about Reproducing ... 0 votes 1 answer 192 views ### Green function of the triangular kernel? What is the green function of the triangular kernel$K$: $$K(x,y)=1-|x-y|$$ where$x,y\in R$such that$|x-y|<1$? 4 votes 0 answers 200 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: ... 6 votes 0 answers 95 views ### Irreducible representations and Jaccard Kernel for Groups? Since each group$G$can be considered as a subgroup of the symmetric group, we might see the elements of$G$as permutations$\pi$. Consider for each$\pi \in G$the set: $$X(\pi) := \{ (i,\pi(i)) |... 6 votes 0 answers 344 views ### Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities? Consider the Frey-Hellegouarch curve given a,b natural numbers:$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$Then the discriminant is given by \Delta = \Delta(a,b) = 16 \left(\frac{ab(a+... 1 vote 1 answer 176 views ### Does kernel regression preserve monotonicity? Consider the Kernel regression estimator:$$\hat{y}(x)=\frac{\sum_{i=1}^n{K(x-x_i)y_i}}{\sum_{i=1}^n{K(x-x_i)}},$$where x,x_1,\dots,x_n\in\mathbb{R}^d, y_1,\dots,y_n\in\mathbb{R}, where K:\... 6 votes 1 answer 289 views ### The abc-conjecture over the positive rationals and Levy-Schoenberg kernels? I am continuing the "abc-adventure" and have a specific question, which needs some explanation: In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3). Consider the ... 7 votes 0 answers 107 views ### Positive definite kernels on categories I'm wondering if there is any work on studying positive definite kernels on (the objects of a) category. By this I mean for a category \mathcal{C}, find a function$$ K: Ob\mathcal{C} \times Ob\... 2 votes 1 answer 190 views ### Connection between non-constant completely monotone function and strictly positive definite kernels (Schoenberg characterization) I'm reading this book chapter, where they stated two alternative characterizations of completely monotone functions$\phi$using (1) Laplace transform of a finite, non-negative Borel measure and also ... 21 votes 1 answer 1k views ### The abc-conjecture as an inequality for inner-products? The abc-conjecture is: For every$\epsilon > 0$there exists$K_{\epsilon}$such that for all natural numbers$a \neq b$we have: $$\frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\... 16 votes 1 answer 772 views ### Are primes linearly separable? Let X_1,\cdots,X_n be finite subsets of some set Z. Then the symmetric difference metric space:$$d(X_i,X_j) = \sqrt{ |X_i|+|X_j|-2|X_i\cap X_j|}$$can be embedded in Euclidean space. The value |... 1 vote 0 answers 136 views ### Karhunen-Loeve expansion of vector valued random processes This thesis (https://www.semanticscholar.org/paper/Karhunen-Loeve-expansions-and-their-applications-Wang/f173dfb99ec4cbd08e779923770466cf1ef3f138) introduces multivariate KL expansion using a ... 4 votes 0 answers 534 views ### Norms of the Dirichlet kernel I guess that the following estimates are classical. Let D_N be the 1D Dirichlet kernel,$$ D_N(t)=\frac{\sin((N+\frac12)t)}{\sin (t/2)}.$$We have for$1<p<\infty, \begin{align} \Vert D_N\... 3 votes 1 answer 77 views ### Different type of measurability of transition kernel Let(E,d)$be a Polish space equipped with the Borel$\sigma$-algebra$\mathcal{E}$. Let$\mathcal{P}(E)$be the space of all probability measures on$(E,\mathcal{E})$. We eqiup this space with the ... 0 votes 1 answer 128 views ### Does the set of positive definite kernels on some set X form a ring? Given some non-empty set$X$, does the set of positive definite kernels on$K_X$form a ring with pointwise addition and multiplication. I am convinced it does not as surely if$k \in K_X$then we ... 3 votes 0 answers 257 views ### Simple (?) question 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 Daumé III. I believe the author fully ... 1 vote 0 answers 123 views ### Preserving the strictly total positivity of special bases by using radial basis functions Let$\left[\alpha,\beta\right]$be a non-empty interval and consider an$\left(n+1\right)$-dimensional subspace$\mathbb{S}_{n}$of$C^n\left(\left[\alpha,\beta\right]\right)$such that$\mathbb{S}_{n}...
Suppose you have two PSD kernels $k(x,y)$ and $k'(x,y)$. Let $\mathcal{H}_k(\mathcal{D})$ and $\mathcal{H}_{k'}(\mathcal{D})$ be the corresponding RKHS's. Now, if we know that for all \$f \in \...