Timeline for Inner product of the spherical cap and Gaussian
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 6 at 3:05 | vote | accept | MMH | ||
| Mar 6 at 3:05 | |||||
| Mar 5 at 0:13 | comment | added | Christian Remling | To elaborate on Iosif's comment, $E\|\eta\|^2=d$, so $E\|\eta\|<\sqrt{d}$ by Cauchy-Schwarz, without a calculation (clearly equality is impossible here), and the exact value could also easily be found (in terms of $\Gamma$ functions) by using the $\chi^2(d)$-distribution. | |
| Mar 4 at 21:48 | comment | added | Iosif Pinelis | Just a very small point: $E\|\eta\|$ is asymptotic, but not equal, to $\sqrt d$. | |
| Mar 4 at 18:27 | vote | accept | MMH | ||
| Mar 4 at 21:23 | |||||
| Mar 4 at 18:02 | comment | added | Kostya_I | @PierrePC, ouch, thanks! I changed $\theta_{Kostya}$ to $\nu$. | |
| Mar 4 at 18:00 | history | edited | Kostya_I | CC BY-SA 4.0 |
added 39 characters in body
|
| Mar 4 at 16:58 | comment | added | Pierre PC | For all $\delta>0$, there exists $C>0$ such that for all $\theta_\mathrm{MMH}\sqrt d\geq\delta$, the expectation is bounded above by $C\sin(\theta_{Kostya})\sqrt d$ and below by $C\sin(\theta_{Kostya})\sqrt d$. So $\theta_\mathrm{MMH}$ can go to zero or to one and we still have a sharp bound, provided the dimension does to infinity fast enough. | |
| Mar 4 at 16:54 | comment | added | MMH | @PierrePC. I see. you are right. I am not sure I understand your second part of your comment. | |
| Mar 4 at 16:53 | vote | accept | MMH | ||
| Mar 4 at 16:54 | |||||
| Mar 4 at 16:41 | comment | added | Pierre PC | I think $\theta_\mathrm{MMH}=\cos(\theta_\mathrm{Kostya})$. I just want to add that this looks like it holds for $\theta$ fixed, but we will still have inequalities up to constants in both directions over a domain where $\theta_\mathrm{MMH}\sqrt d$ is uniformly bounded below. | |
| Mar 4 at 16:32 | history | answered | Kostya_I | CC BY-SA 4.0 |