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.

翻译：本文研究了矩阵分解与因子分解机的等式型Clarke次微分链式法则。具体而言，我们证明对于这些问题，当潜在维度大于问题规模的某个倍数（即轻微过参数化）且损失函数局部Lipschitz连续时，次微分链式法则处处成立。此外，我们通过若干有趣的构造分析了证明的紧致性，并从优化视角得出若干重要观察结论；例如，我们证明对于所有此类问题，计算稳定点是平凡的。文中也讨论了张量推广与神经网络的扩展，尽管这些问题大多仍未解决。