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Questions tagged [epsilon-nets]

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2
votes
1answer
62 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 ...
1
vote
0answers
129 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 ...
1
vote
0answers
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 $S_\text{hit}$ ...
8
votes
1answer
307 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 ...
4
votes
0answers
248 views

Open problems on epsilon nets

What would be a good source for open problems for (weak) epsilon nets? Is there a good survey/article that summarizes the recent advancements on the topic?
10
votes
1answer
800 views

$\epsilon$-nets with respect to the cut norm

The cut norm $||A||_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
7
votes
2answers
220 views

Counting Metrics

Say that I have a set of $n$ points $N$, and am interested in metrics $d:N\times N \rightarrow \mathbb{R}$ over $N$. Let $M$ denote the set of all metrics over $N$. Now let me define the distance ...
10
votes
2answers
400 views

Consequences of lower bounds for $\epsilon$-nets on approximation

Many here are probably aware of Alon's recent super-linear lower bounds for $\epsilon$-nets in a natural geometric setting [PDF]. I would like to know what, if anything, such a lower bound implies ...