All Questions
10 questions with no upvoted or accepted answers
4
votes
0
answers
156
views
Known relations between mutual information and covering number?
This is a question about statistical learning theory. Consider a hypothesis class $\mathcal{F}$, parameterized by real vectors $w \in \mathbb{R}^p$. Suppose I have a data distribution $D \sim \mu$ and ...
- 141
4
votes
0
answers
164
views
Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
2
votes
0
answers
448
views
Relation between pseudo-dimension and Rademacher complexity
With techniques of Dudley's entropy bound and Haussler's upper bound one can show that there exists a constant $C$ such that any class of $\{0,1\}$ indicator functions with Vapnik-Chervonenkis ...
- 517
1
vote
0
answers
24
views
Minimax statistical estimation of proximal transform $\mbox{prox}_g(\theta_0)$, from linear model data $y_i := x_i^\top \theta_0 + \epsilon_i$
tl;dr: My question pertains the subject of minimax estimation theory (mathematical statistics), in the context of linear regression.
Given a vector $\theta_0 \in \mathbb R^d$, consider the linear ...
- 6,853
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
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
1
vote
0
answers
99
views
Covering number after projection
In these lecture notes on Statistical Learning Theory we find the following definitions for covering numbers:
Definition. Let $(\mathcal{W}, d)$ be a metric space and $\mathcal{F} \subset \mathcal{W}$...
0
votes
0
answers
97
views
Verification of a certain computation of VC dimension
Disclaimer: I'm not very familiar with the concept of VC dimensions and how to manipulate such objects. I'd be grateful if expects on the subject (learning theory, probability), could kindly proof ...
- 6,853
0
votes
0
answers
195
views
Upper-bound for bracketing number in terms of VC-dimension
Let $P$ be a probability distribution on a measurable space $\mathcal X$ (e.g;, some euclidean $\mathbb R^m$) and let $F$ be a class of funciton $f:\mathcal X \to \mathbb R$. Given, $f_1,f_2 \in F$, ...
- 6,853