Model Selection for Non-Negative Tensor Factorization with Minimum Description Length

Non-negative tensor factorization (NTF) is a widely used multi-way analysis approach that factorizes a high-order non-negative data tensor into several non-negative factor matrices. In NTF, the non-negative rank has to be predetermined to specify the model and it greatly influences the factorized matrices. However, its value is conventionally determined by specialists' insights or trial and error. This paper proposes a novel rank selection criterion for NTF on the basis of the minimum description length (MDL) principle. Our methodology is unique in that (1) we apply the MDL principle on tensor slices to overcome a problem caused by the imbalance between the number of elements in a data tensor and that in factor matrices, and (2) we employ the normalized maximum likelihood (NML) code-length for histogram densities. We employ synthetic and real data to empirically demonstrate that our method outperforms other criteria in terms of accuracies for estimating true ranks and for completing missing values. We further show that our method can produce ranks suitable for knowledge discovery.