On subdifferential chain rule of matrix factorization and beyond

In this paper, we study equality-type Clarke subdifferential chain rules of matrix factorization and factorization machine. Specifically, we show for these problems that provided the latent dimension is larger than some multiple of the problem size (i.e., slightly overparameterized) and the loss function is locally Lipschitz, the subdifferential chain rules hold everywhere. In addition, we examine the tightness of the analysis through some interesting constructions and make some important observations from the perspective of optimization; e.g., we show that for all this type of problems, computing a stationary point is trivial. Some tensor generalizations and neural extensions are also discussed, albeit they remain mostly open.