Questions tagged [epsilon-nets]

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8 questions
77 views

Empirical Rademacher averages versus Hoeffdings bound

Let $M$ be finite set with $n$ distinct elements. I want to probalistically approximate the relative counts $\frac{|P(Q)|}{|M|}$ of $Q \subseteq M$, where $P(Q) = |P \cap M|$. An upper-bound for ...
133 views

Clarification needed on an algorithm for $\epsilon$-net construction for the column space of PSD matrices

I found an algorithm for constructing an $\epsilon$-net for a positive semidefine matrix $A\in[-1,1]^{n\times n}$ which has $rank(A)=d$, described in the paper The approximate rank of a matrix and ...
142 views

Hitting sets for sets of VC dimension d

Let $S$ be a collection of sets of binary vectors (in $\{0,1\}^m$) $S_1, S_2, \dotsc, S_t$ (say $t = O(m^d)$) each of VC dimension $d$. What can be said about the size of a hitting set $S_\text{hit}$ ...
315 views

How intrinsic is the $d^d$ term in the running time for constructing $\varepsilon$-nets in range spaces of VC-dimension d?

An $\varepsilon$-net for a range space $(X,\mathcal{R})$ is a subset $N$ of $X$ such that $N\cap R$ is nonempty for all $R\in \mathcal{R}$ such that $|X\cap R| \ge \varepsilon |X|$. Given a range ...