3.3. Revised NCA
The work in [27] also focuses on enhancing the NCA criteria. The work in [27] proposed revised NCA (NCAr), where the third criterion of NCA is revised to improve the applicability of NCA. As discussed earlier, to ensure a unique solution for the matrix factorization problem, the third criterion of NCA requires the matrix S to have full row rank, which implies that the number of TFs must be less than or equal to the number of experiments. This requirement significantly limits the sample size of TFs. The work in [27] revises the third criterion of NCA based on the observation that most of the genes are only regulated by a smaller number of TFs than the total number of TFs (i.e., the connectivity matrix A is row-wise sparse). In particular, this condition, instead of being associated with the rank properties of matrix S, is related to the rank properties of reduced-size matrices. Particularly, it requires that the number of experiments for each gene be greater than or equal to the number of TFs regulating that gene. The revised criterion enables NCA to be applicable to a wider class of TRN inference problems, since the number of TFs regulating a gene is generally less than five or six [27]. In this way, a large dimensional regulatory network can be uniquely inferred, even in the presence of a limited number of experiments.