The reason why I am using random projections in the latest test is because I am testing an algorithm that iteratively calculates the inverse Cholesky factor of the covariance matrix and am testing it on Mnist images. The cov matrices made from raw Mnist images are non-invertible, but projecting them to a much smaller dimension allows me to actually test the algorithm on non-synthetic data.

I do not actually need more than I have, but I'll keep your link in mind if I ever need random projections though.