I found this result in a research paper. not sure how to derive it. any help?? suppose an n dimensional unit vector v=(v1,v2,...,vn) is chosen uniformly at random from an n dimensional unit sphere. clearly v1^2+v2^2+...+vn^2=1-----------------(1) Then for any delta>0, with probability at least (1-delta) , we have v1^2>=1/(n*log(1/delta))
This means that we have to show that the probability, P[v1^2<1/ (n*log(1/delta))] is bounded by delta
I don't know how to show this but what I can see is that there exists one i, such that vi^2>=1/n ...
In article <63a9e35e-5233-402b-969f-e249c24a2...@k37g2000hsf.googlegroups.com>,
kaushik.sinha...@gmail.com wrote: > I found this result in a research paper. not sure how to derive it. > any help?? > suppose an n dimensional unit vector v=(v1,v2,...,vn) is chosen > uniformly at random from an n dimensional unit sphere. clearly > v1^2+v2^2+...+vn^2=1-----------------(1) > Then for any delta>0, with probability at least (1-delta) , we have > v1^2>=1/(n*log(1/delta))
That can't be true for all d > 0. For example, let d = e^(-1/n). Then you are saying v1^2 >= 1 with probability >= 1 - e^(-1/n).
> In article > <63a9e35e-5233-402b-969f-e249c24a2...@k37g2000hsf.googlegroups.com>,
> kaushik.sinha...@gmail.com wrote: > > I found this result in a research paper. not sure how to derive it. > > any help?? > > suppose an n dimensional unit vector v=(v1,v2,...,vn) is chosen > > uniformly at random from an n dimensional unit sphere. clearly > > v1^2+v2^2+...+vn^2=1-----------------(1) > > Then for any delta>0, with probability at least (1-delta) , we have > > v1^2>=1/(n*log(1/delta))
> That can't be true for all d > 0. For example, let d = e^(-1/n). Then > you are saying v1^2 >= 1 with probability >= 1 - e^(-1/n).
> > This means that we have to show that the probability, P[v1^2<1/ > > (n*log(1/delta))] is bounded by delta
> > I don't know how to show this but what I can see is that there exists > > one i, such that vi^2>=1/n ...- Hide quoted text -
kaushik.sinha...@gmail.com wrote: > On Jul 23, 10:23 pm, The World Wide Wade <aderamey.a...@comcast.net> > wrote: > > In article > > <63a9e35e-5233-402b-969f-e249c24a2...@k37g2000hsf.googlegroups.com>,
> > kaushik.sinha...@gmail.com wrote: > > > I found this result in a research paper. not sure how to derive it. > > > any help?? > > > suppose an n dimensional unit vector v=(v1,v2,...,vn) is chosen > > > uniformly at random from an n dimensional unit sphere. clearly > > > v1^2+v2^2+...+vn^2=1-----------------(1) > > > Then for any delta>0, with probability at least (1-delta) , we have > > > v1^2>=1/(n*log(1/delta))
> > That can't be true for all d > 0. For example, let d = e^(-1/n). Then > > you are saying v1^2 >= 1 with probability >= 1 - e^(-1/n).
> > > This means that we have to show that the probability, P[v1^2<1/ > > > (n*log(1/delta))] is bounded by delta
> > > I don't know how to show this but what I can see is that there exists > > > one i, such that vi^2>=1/n ...- Hide quoted text -
> > - Show quoted text -
> restrict delta to 0 <delta < 1/2
Why, is this in the paper? Where does 1/2 come from?
In article <63a9e35e-5233-402b-969f-e249c24a2...@k37g2000hsf.googlegroups.com>,
kaushik.sinha...@gmail.com wrote: > I found this result in a research paper. not sure how to derive it. > any help?? > suppose an n dimensional unit vector v=(v1,v2,...,vn) is chosen > uniformly at random from an n dimensional unit sphere.
Do you mean an (n+1)-dimensional unit vector from an n-dimensional unit sphere?
