Hi
I have a problem understanding why the normal distribution is a good approximation of the uniform distribution in a sphere. I would expect the probability to be higher in the center with a normal distribution. Wouldnt this alpha way smaller than it would be with a uniform distribution?
Is it possible that you post the notes from the video on github?
A Gaussian vector has a spherical distribution, so if you forget about the norm of the vector you see that it is a good approximation of the uniform distribution. It remains the problem of the norm. But fortunately the norm of a Gaussian vector in high dimension is highly concentrated around its mean which is \(O(\sqrt{d})\). Therefore you see that if you turn a blind eyes on its fluctuation around its mean, you see that the two distributions are very close and behave similarly.
"I would expect the probability to be higher in the center with a normal distribution. "
I am not sure to follow which center you are talking about. The probability to have a point around the center of the ball is awfully small. You can do the computation if you want and you will see it will go to zero with d going to infinity. You can also do some simulations to see this approximation is totally legit.
Proof Spherical cap distribution
Hi
I have a problem understanding why the normal distribution is a good approximation of the uniform distribution in a sphere. I would expect the probability to be higher in the center with a normal distribution. Wouldnt this alpha way smaller than it would be with a uniform distribution?
Is it possible that you post the notes from the video on github?
Best Silvio
Hello Silvio,
The notes are already on GitHub (lecture 9c).
A Gaussian vector has a spherical distribution, so if you forget about the norm of the vector you see that it is a good approximation of the uniform distribution. It remains the problem of the norm. But fortunately the norm of a Gaussian vector in high dimension is highly concentrated around its mean which is \(O(\sqrt{d})\). Therefore you see that if you turn a blind eyes on its fluctuation around its mean, you see that the two distributions are very close and behave similarly.
"I would expect the probability to be higher in the center with a normal distribution. "
I am not sure to follow which center you are talking about. The probability to have a point around the center of the ball is awfully small. You can do the computation if you want and you will see it will go to zero with d going to infinity. You can also do some simulations to see this approximation is totally legit.
Cheers,
Nicolas
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