Sparse and Simple Structure Estimation via Prenet Penalization

We propose a prenet (product-based elastic net), a novel penalization method for factor analysis models. The penalty is based on the product of a pair of elements in each row of the loading matrix. The prenet not only shrinks some of the factor loadings toward exactly zero but also enhances the simplicity of the loading matrix, which plays an important role in the interpretation of the common factors. In particular, with a large amount of prenet penalization, the estimated loading matrix possesses a perfect simple structure, which is known as a desirable structure in terms of the simplicity of the loading matrix. Furthermore, the perfect simple structure estimation via the proposed penalization turns out to be a generalization of the k-means clustering of variables. On the other hand, a mild amount of the penalization approximates a loading matrix estimated by the quartimin rotation, one of the most commonly used oblique rotation techniques. Simulation studies compare the performance of our proposed penalization with that of existing methods under a variety of settings. The usefulness of the perfect simple structure estimation via our proposed procedure is presented through various real data applications. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11336-022-09868-4.