Random projections became popular tools to process big data. In particular, when applied to Nonnegative Matrix Factorization (NMF), it was shown that structured random projections were far more efficient than classical strategies based on Gaussian compression. However, they remain costly and might not fully benefit from recent fast random projection techniques. In this paper, we thus investigate an alternative to structured random projections-named Gaussian compression stream-which (i) is based on Gaussian compressions only, (ii) can benefit from the above fast techniques, and (iii) is shown to be well-suited to NMF.