Sampling from high dimensional sphere with noise

I want to generate a sample of vectors from the high dimensional sphere with noise.

I.E. I'm trying to create a sample such that any vector X is in R^N and holds ||X+epsilon||^2 = 1 where epsilon is iid vector in R^N of which any component epsilon_j is distributed N(0,sigma^2).

Does someone have any idea how to implement it? I'd prefer to use R.

Thank you!

Solution

I think this should work. It could easily be turned into a function.

d = 5         # number of dimensions
n_draws = 100 # number of draws
sigma = 0.2   # standard deviation

I begin by sampling random vectors that should be uniformly distributed on the unit sphere. I do this by normalizing draws from a d-dimensional multivariate normal distribution. (There's probably a more direct way to do this step, but I'll be using rmvnorm again later so this was convenient.) I call them dirs because, since we're normalizing, all we're really doing in this step is sampling "directions".

library(mvtnorm)
# sample
dirs = rmvnorm(n = n_draws, mean = rep(0, d))
# normalize
dirs = dirs / sqrt(rowSums(dirs^2))

Now we do another draw from the multivariate normal to add noise.

x = dirs + rmvnorm(n = n_draws, mean = rep(0, d), sigma = sigma * diag(d))

To map this to the variables you used in your question, define Y = X + epsilon. My dirs is Y, then the noise I add is -epsilon; adding them yields the X that you asked for.