This article presents a form of bi-cross-validation (BCV) for choosing the rank in outer product models, especially the singular value decomposition (SVD) and the nonnegative matrix factorization (NMF). Instead of leaving out a set of rows of the data matrix, we leave out a set of rows and a set of columns, and then predict the left out entries by low rank operations on the retained data. We prove a self-consistency result expressing the prediction error as a residual from a low rank approximation. Random matrix theory and some empirical results suggest that smaller hold-out sets lead to more over-fitting, while larger ones are more prone to under-fitting. In simulated examples we find that a method leaving out half the rows and half the columns performs well.
- Pub Date:
- August 2009
- Statistics - Applications
- Published in at http://dx.doi.org/10.1214/08-AOAS227 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)