Konstantin Siroki and Korbinian Strimmer’s contribution to the Discussion of ‘Vintage factor analysis with varimax performs statistical inference’ by Rohe and Zeng

原文摘要

We congratulate Rohe and Zeng on their interesting paper and would like to share two comments and propose one question for their consideration. First, it is impressive to see the large variety of models addressed by the ‘vintage sparse PCA’ (vsp) framework, ranging from traditional factor models to explainable and sparse PCA but also including topic models and stochastic block models. In our view, the demonstration of the broad applicability of the ‘compress and rotate’ strategy is a major contribution of this article. Unfortunately, much of this material is relegated to the Appendix of the article. This extends earlier discussions of the benefits of rotating and sparsifying principal components in more classical use cases, e.g. by Jolliffe (2002, Chapter 11) and by Trendafilov and Adachi (2015). Second, the problem of identifiability of loadings and factors in factor analysis is mirrored in the problem of identifying optimal whitening transformations (Jendoubi & Strimmer, 2019; Kessy et al., 2018). There are infinitely many such transformations, and just like in factor analysis it is only the constraints on the covariance-based or correlation-based loadings that allow to distinguish among and select corresponding whitening methods (to include PCA-based whitening, ZCA whitening, and Cholesky whitening). Crucially, PCA-based whitening is itself already the result of a rotation, so vsp may be interpreted as two consecutive rotations, not just one, the first leading to optimal compression and the second to sparsify the loadings of the top-ranking factors. This leads to our main question. It is shown by Rohe and Zeng that under the leptokurtic distributional assumption the varimax rotation is able to identify the original underlying factors. Now, varimax that lies at the heart of vsp is part of the larger orthomax family (Harman, 1976) which comprises not only varimax, but quartimax, biquartimax, equamax, and parsimax, and in fact many other types of rotations. In turn, the orthomax family is itself a special case of even richer families (e.g. Browne, 2001) that also include oblique transformations. A result by Bernaards and Jennrich (2003) suggests that all members of the orthomax family allow to recover simple structure, not just varimax. The key argument in Bernaards and Jennrich (2003) is essentially the same as the motivation put forward by Rohe and Zeng about varimax and sparsity. Hence, we ask whether the leptokurtic condition is strict enough and sufficient to single out varimax or whether perhaps the whole orthomax family may be compatible with it?

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