  iALS is a popular algorithm for learning matrix factorization models from
implicit feedback with alternating least squares. This algorithm was invented
over a decade ago but still shows competitive quality compared to recent
approaches like VAE, EASE, SLIM, or NCF. Due to a computational trick that
avoids negative sampling, iALS is very efficient especially for large item
catalogues. However, iALS does not scale well with large embedding dimensions,
d, due to its cubic runtime dependency on d. Coordinate descent variations,
iCD, have been proposed to lower the complexity to quadratic in d. In this
work, we show that iCD approaches are not well suited for modern processors and
can be an order of magnitude slower than a careful iALS implementation for
small to mid scale embedding sizes (d ~ 100) and only perform better than iALS
on large embeddings d ~ 1000. We propose a new solver iALS++ that combines the
advantages of iALS in terms of vector processing with a low computational
complexity as in iCD. iALS++ is an order of magnitude faster than iCD both for
small and large embedding dimensions. It can solve benchmark problems like
Movielens 20M or Million Song Dataset even for 1000 dimensional embedding
vectors in a few minutes.
