All Questions
11 questions
1
vote
0
answers
144
views
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 $L^2(\pi)$ ...
- 111
1
vote
0
answers
417
views
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 ...
- 6,853
1
vote
0
answers
100
views
$L_1$ 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 $...
- 6,853
1
vote
0
answers
63
views
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
$$
\...
- 6,853
3
votes
0
answers
461
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) := ...
- 6,853
1
vote
0
answers
233
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 ...
- 6,853
3
votes
1
answer
688
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 ...
- 41
5
votes
1
answer
146
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 ...
- 931
0
votes
1
answer
217
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). ...
- 11
3
votes
1
answer
870
views
Karhunen-Loeve expansion for discrete-time process
Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...
user24451
3
votes
0
answers
236
views
Is every covariance operator the covariance of a measure?
Let $X$ be a topological linear space over $\mathbb R$ which is complete and Hausdorff with a dual space that separates points. Let $k : X^* \to X$ be an arbitrary covariance operator. i.e., any ...
- 8,512