Recovering Hidden Diagonal Structures via Non-Negative Matrix Factorization with Multiple Constraints

Revealing data with intrinsically diagonal block structures is particularly useful for analyzing groups of highly correlated variables. Earlier researches based on non-negative matrix factorization (NMF) have been shown to be effective in representing such data by decomposing the observed data into two factors, where one factor is considered to be the feature and the other the expansion loading from a linear algebra perspective. If the data are sampled from multiple independent subspaces, the loading factor would possess a diagonal structure under an ideal matrix decomposition. However, the standard NMF method and its variants have not been reported to exploit this type of data via direct estimation. To address this issue, a non-negative matrix factorization with multiple constraints model is proposed in this paper. The constraints include an sparsity norm on the feature matrix and a total variational norm on each column of the loading matrix. The proposed model is shown to be capable of efficiently recovering diagonal block structures hidden in observed samples. An efficient numerical algorithm using the alternating direction method of multipliers model is proposed for optimizing the new model. Compared with several benchmark models, the proposed method performs robustly and effectively for simulated and real biological data.