Bi-cross-validation of the SVD and the nonnegative matrix factorization
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.
Optimality, computation, and interpretation of nonnegative matrix factorizations
The notion of low rank approximations arises from many important applications. When the low rank data are further required to comprise nonnegative values only, the approach by nonnegative matrix factorization is particularly appealing. This paper intends to bring about three points. First, the theoretical Kuhn-Tucker optimality condition is described in explicit form. Secondly, a number of numerical techniques, old and new, are suggested for the nonnegative matrix factorization problems. Thirdly, the techniques are employed to two real-world applications to demonstrate the difficulty in interpreting the factorizations.
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