  We propose a $prenet$ ($pr$oduct $e$lastic $net$), which is a new
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 prenet 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. Thus, the proposed penalty bridges a gap
between the perfect simple structure and the quartimin rotation. Monte Carlo
simulation is conducted to investigate the performance of the proposed
procedure. Three real data analyses are given to illustrate the usefulness of
our penalty.
