Tagged Questions
2
votes
1answer
73 views
It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?
This question is motivated from a previous question, but is in itself independent of it.
So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
0
votes
1answer
51 views
Show that $C([0,1],\mathbb{R})$ with the $L_2$ inner product norm is not a Hilbert space.
I need to prove that all continuous functions on the closed set $[0,1]$ is not a Hilbert space. Given the $L_2$ norm.
I guess I need to show that every Cauchy sequence in the space, does not ...
0
votes
0answers
41 views
Representer theorem RKHS?
$\displaystyle\min_{f\in H_k} \gamma ||f||_{K}^2+f^TLf$
Given a RKHS $H_k$ does the representer theorem hold for this minimization problem over the function $f$? Here, $L$ is a fixed p.s.d ...
1
vote
0answers
71 views
Reproducing Kernels are Positive Definite. Does the converse hold true?
Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
2
votes
1answer
132 views
Notation: Representer Theorem for Reproducing kernel hilbert spaces
Am studying the basic concepts of RKHS and the representer theorem:
In
$f(x_i)=<f,k(x_i,\mathbb{.})>$, what does $ f$ on the r.h.s denote? What is its structure-is it a vector? I was thinking ...
1
vote
1answer
98 views
Countable Hilbert Spaces
I have seen a simple proof that no banach space over $\mathbb{R}$ can be of countably infinite dimension. However since the space of all square integrable functions on the unit interval forms a ...
0
votes
0answers
259 views
Proof of Isometry: Inner Product Preserving Map
For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
2
votes
1answer
135 views
A question on norm of error vector
Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
