Entrywise splitting cross-validation in generalized factor models: from sample splitting to entrywise splitting

The generalized factor models have been widely employed for dimension reduction across various types of multivariate data, including binary choices, counts, and continuous observations. While determining the number of factors in such models has received significant scholarly attention, it remains an open challenge in the field. In this paper, we propose a cross-validation (CV) method based on entrywise splitting (ES), rather than sample splitting, to address this problem. Similar to traditional cross-validation, this approach primarily prevents the underestimation of the number of factors. We then introduce a penalized entrywise splitting cross-validation criterion, which integrates the original CV with information theoretic criteria by adding a penalty term. Its consistency is established under mild conditions in a high-dimensional setting, where both the sample size and the number of features grow to infinity. Furthermore, we extend our methodology to random missing data with different probability scenarios. We evaluate the performance of the proposed method through comprehensive simulations and apply it to a mouse brain single-cell RNA sequencing dataset.