All Questions
14 questions
4
votes
1
answer
1k
views
Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition
I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in this paper .
Among the bibliography I'm using there are "On the mathematical fundamentals of learning" and "...
- 151
4
votes
0
answers
111
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,...
- 316
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
495
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 ...
- 163
2
votes
3
answers
866
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 ...
- 221
2
votes
1
answer
145
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 $K$ ...
- 5,405
2
votes
0
answers
73
views
RKHS lying in another RKHS
Suppose $H_1$ and $H_2$ are reproducing kernel Hilbert spaces such that $H_1 \subset H_2$. For $f \in H_1$, when can I bound $\|f \|_1$ with $C\|f\|_2$ (for some $C$)? Is there a relationship between ...
- 93
2
votes
0
answers
553
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 $\...
- 121
2
votes
1
answer
959
views
Do kernels provide a basis for a RKHS?
Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, \...
- 461
1
vote
0
answers
83
views
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) ...
- 43
1
vote
0
answers
133
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 ...
- 93
0
votes
1
answer
736
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 $\hat{k}$ ...
- 13
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
0
votes
1
answer
517
views
Injective inclusion map from RKHS function space to $L_p(\mu)$
Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$.
At a certain part in a proof I ...
- 103