Timeline for Use covering number to get uniform concentration from pointwise concentration
Current License: CC BY-SA 4.0
11 events
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| Feb 3, 2019 at 12:40 | comment | added | dohmatob | OK, i just checked and it seems this is indeed not how people usually define it en.wikipedia.org/wiki/Covering_number#Definition. Here is my working definition: Given $\epsilon \ge 0$, a collection $C_\epsilon$ of subsets of $\Theta$ is said to be an $\epsilon$-cover for $\Theta$ if the diameter of each $C \in \mathcal C_\epsilon$ is at most $2\epsilon$ and $\Theta \subseteq \bigsqcup_{C \in C_\epsilon} C $. You will agree with me that this is essentially equivalent to the standard one ;) | |
| Feb 3, 2019 at 12:36 | comment | added | Aryeh Kontorovich | That's not the standard definition of $\epsilon$-cover. | |
| Feb 3, 2019 at 12:35 | comment | added | dohmatob | $\mathcal C_\epsilon$ is a collection of subsets of $\Theta$. Each element / point $C$ of $\mathcal C_\epsilon$ is a subset of $\Theta$. | |
| Feb 3, 2019 at 11:59 | comment | added | Aryeh Kontorovich | What is the "type" of $\mathcal{C}_\epsilon$? Is it a subset of $\Theta$? | |
| Feb 3, 2019 at 11:28 | comment | added | dohmatob | I don't get that. I'm not interested by the distance about two points in $C_\epsilon$ (which wouldn't make much sense, since there is no geometry on $\mathcal C_\epsilon$, just a set of things...). I'm interested in the distance between two points $x$ and $y$ with $x, y \in C \in \mathcal C_\epsilon \subseteq \mathcal 2^\Theta$, and this can certainly be talked about, since $C$ is a subset of the metric space $\Theta$. No ? | |
| Feb 2, 2019 at 20:23 | comment | added | Aryeh Kontorovich | Actually $\mathcal{C}_\epsilon$ is a cover (of "type" set) and $C\in \mathcal{C}_\epsilon$ is a point -- so we should be talking about the distance between any two points in $\mathcal{C}_\epsilon$. The latter could be large -- roughly as large as the diameter of $\Theta$. | |
| Feb 1, 2019 at 16:12 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 1, 2019 at 16:11 | comment | added | dohmatob | That's a typo / thinko. I actually meant $2\epsilon$. Thanks. | |
| Feb 1, 2019 at 12:39 | comment | added | Aryeh Kontorovich | It's not true that the distance of any two points in $C$ is at most $\epsilon$. | |
| Feb 1, 2019 at 7:00 | history | edited | dohmatob | CC BY-SA 4.0 |
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| Feb 1, 2019 at 6:54 | history | answered | dohmatob | CC BY-SA 4.0 |