Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization

Abstract

Non-negative 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 non-negative. 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 well-known 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 Laplace-PCA method might be best. However, when the orthogonality of the underlying components is unknown, none of the methods seemed preferable.

Inferring therapeutic vulnerability within tumors through integration of pan-cancer cell line and single-cell transcriptomic profiles

“…When coupled with CCA, both regression models gave comparable results. Additionally, while the selection of the optimal k in either CCA or NMF plays a central role in algorithm performance 30,38 , given that the computational goal is to model drug response, we implemented feature selection on post-integration embeddings to include subspace features that correlate with drug response (see Methods). Through this, scIDUC was able to quickly select only a few meaningful features for model training and prediction without searching for optimal inner dimensions in an unsupervised fashion.…”

Scalable joint non-negative matrix factorisation for paired single cell gene expression and chromatin accessibility data

“…The first relies on a modified version of BIC for NMF [35], given below: where denotes the NMF reconstructed matrix of a given modality. Alternatively, intNMF uses kneedle [36] to identify where the reduction in loss flattens with the addition of further topics.…”

“…The number of topics is then selected based on either Bayesian information criterion (BIC) or the knee point of the loss as function of topic number. The first relies on a modified version of BIC for NMF [35], given below:…”

Functional transcriptional signatures for tumor-type-agnostic phenotype prediction

“…This models the unknown number of archetypes as a latent variable that can be directly optimized over. This method was selected due to its simplicity to implement and its superior performance compared to multiple other methods for selecting the number of factors in NMF [62]. For N-NMF, the minimum number of factors was set as k = 2 (due to the normalization constraint), and the maximum (a requirement of the approach) was chosen to be k = 30.…”

“…This approach models the unknown number of archetypes as a latent variable that can be directly optimized over. This method was selected due to its simplicity and superior performance compared to alternatives [30]. When training the archetype model on normal tissue transcriptomic data (GTEx), the minimum number of factors was set to k = 2 (due to the normalization constraint), with the maximum set at k = 30.…”