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Tagged with learning-theory kernels
8
questions
2
votes
1
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57
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Non-linear transforms of RKHS question
I was reading the paper Norm Inequalities in Nonlinear Transforms (referenced in this question) but ran into difficulties, so I was wondering if anyone could help?
I think I follow the paper until I ...
- 41
1
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0
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368
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Conditions for equivalence of RKHS norm and $L^2(P)$ norm
Let $K$ be a psd kernel on an abstract space $X$ and let $H_K$ be the induced Reproducing Kernel Hilbert Space (RKHS). Let $P$ be a probability measure on $X$ such that $H_K \subseteq L^2(P_X)$ and ...
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1
vote
0
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95
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$L_1$ convergence rates for multivariate kernel density estimation
Let $X$ be a random variable on $\mathbb R^d$ with probability density function $f$, and let $X_1,\ldots,X_n$ of $X$ be $n$ iid copies of $X$. Given a bandwidth parameter $h=h_n > 0$ and a kernel $...
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1
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0
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56
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Properties of a kernel convolution $K'(x,y) = \int_X\int_X K_0(x,a)K(a,b)K_0(b,y)d\mu(a)d\mu(b)$ where $K$ and $K_0$ are kernels on $(X,\mu)$
Let $(X,\mu)$ be a probability measure space and $K:X \times X \to \mathbb R$ be a (psd) kernel on $X$. Let $K_0$ be another kernel on $X$ and defined a new kernel $\widetilde K$ on $X$ by
$$
\...
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2
votes
1
answer
156
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Representer theorem for a loss / functional of the form $L(h) := \sum_{i=1}^n (|h(x_i)-y_i|+t\|h\|)^2$
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 ...
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6
votes
1
answer
458
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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 ...
- 373
3
votes
0
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427
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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) := ...
- 6,824
1
vote
0
answers
228
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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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