  Dimensionality reduction and matrix factorization techniques are important
and useful machine-learning techniques in many fields. Nonnegative matrix
factorization (NMF) is particularly useful for spectral analysis and image
processing in astronomy. I present the vectorized update rules and an
independent proof of their convergence for NMF with heteroscedastic
measurements and missing data. I release a Python implementation of the rules
and use an optical spectroscopic dataset of extragalactic sources as an example
for demonstration. A future paper will present results of applying the
technique to image processing of planetary disks.
