Today I had a few errands to run but my working time was spend with the distribution from hell (again). Today I gave up and posted to math exchange http://math.stackexchange.com/questions/648577/the-radial-part-of-a-normal-distribution unfortunatly I don't think this will yield an answer.
My thinking so far can be summed to.
What problem is being solved?
The entires of the matrix look like guassian samples, but all rows have the same length. By scaling with $s_i$ this should be fixed. Conclusion: the s behaves like the lenght of a gaussian vector of size d.
It seems intuitive to me that this is related to the unit ball, but I am not fully on page with it still, and especially the weird inversion of A is just killing me.
EDIT: Det der synes at være formålet med S er at få rækkerne i V til at have længder der varierer på samme måde som hvis elementerne i V´ havde været samplet i.i.d fra N(0,1). Jeg tænker at det samme må kunne opnås ved at trække d samples fra en N(0,1) og gemme R^2 normen af disse som indgang i S. Det kræver selvfølgelig D x d samples, men det er en engangsforestilling. << from an email i wrote my supervisor the next day. This is how i ended up doing it: sample d times from a N(0,1) normal distribution and use the length of resulting vectors as entries in S.
My thinking so far can be summed to.
What problem is being solved?
The entires of the matrix look like guassian samples, but all rows have the same length. By scaling with $s_i$ this should be fixed. Conclusion: the s behaves like the lenght of a gaussian vector of size d.
It seems intuitive to me that this is related to the unit ball, but I am not fully on page with it still, and especially the weird inversion of A is just killing me.
EDIT: Det der synes at være formålet med S er at få rækkerne i V til at have længder der varierer på samme måde som hvis elementerne i V´ havde været samplet i.i.d fra N(0,1). Jeg tænker at det samme må kunne opnås ved at trække d samples fra en N(0,1) og gemme R^2 normen af disse som indgang i S. Det kræver selvfølgelig D x d samples, men det er en engangsforestilling. << from an email i wrote my supervisor the next day. This is how i ended up doing it: sample d times from a N(0,1) normal distribution and use the length of resulting vectors as entries in S.
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