Abstract
Nonnegative matrix factorization is a relatively new method of matrix decomposition which factors an m × n data matrix X into an m × k matrix W and a k × n matrix H, so that X ≈ W × H. Importantly, all values in X, W, and H are constrained to be nonnegative. NMF can be used for dimensionality reduction, since the k columns of W can be considered components into which X has been decomposed. The question arises: how does one choose k? In this paper, we first assess methods for estimating k in the context of NMF in synthetic data. Second, we examine the effect of normalization on this estimate’s accuracy in empirical data. In synthetic data with orthogonal underlying components, methods based on PCA and Brunet’s Cophenetic Correlation Coefficient achieved the highest accuracy. When evaluated on a wellknown real dataset, normalization had an unpredictable effect on the estimate. For any given normalization method, the methods for estimating k gave widely varying results. We conclude that when estimating k, it is best not to apply normalization. If the underlying components are known to be orthogonal, then Velicer’s MAP or Minka’s LaplacePCA method might be best. However, when the orthogonality of the underlying components is unknown, none of the methods seemed preferable.
Original language  English 

Article number  2840 
Journal  Mathematics 
Volume  9 
Issue number  22 
DOIs  
State  Published  Nov 1 2021 
Keywords
 Factorization rank
 Highdimensional data
 Nonnegative matrix factorization
 Normalization
 Number of factored components
 PCA
 Unsupervised learning
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Eric A. Ross, PhD, ScM (Director), Karthik Devarajan, PhD (Staff), Yunyun Zhou, PhD (Staff), Yan Zhou, MSE, PhD (Staff), Brian Egleston, PhD, MPP (Staff) & Jill S. Hasler, PhD (Staff)
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