This paper studies the nonsmooth optimization landscape of the $\ell_1$-norm rank-one symmetric matrix factorization problem using tools from second-order variational analysis. Specifically, as the main finding of this paper, we show that any second-order stationary point (and thus local minimizer) of the problem is actually globally optimal. Besides, some other results concerning the landscape of the problem, such as a complete characterization of the set of stationary points, are also developed, which should be interesting in their own rights. Furthermore, with the above theories, we revisit existing results on the generic minimizing behavior of simple algorithms for nonsmooth optimization and showcase the potential risk of their applications to our problem through several examples. Our techniques can potentially be applied to analyze the optimization landscapes of a variety of other more sophisticated nonsmooth learning problems, such as robust low-rank matrix recovery.

翻译：本文利用二阶变分分析工具研究ℓ₁-范数秩一对称矩阵分解问题的非光滑优化几何。具体而言，作为本文的主要发现，我们证明该问题的任意二阶驻点（从而局部极小点）实际上均为全局最优点。此外，我们还得到了关于该问题几何结构的一些其他结果，例如驻点集的完整刻画，这些结果本身亦具有独立价值。进一步地，基于上述理论，我们重新审视了关于非光滑优化中简单算法通用极小化行为的现有结论，并通过若干算例揭示了将其应用于本问题时的潜在风险。我们的技术方法可推广至分析其他更复杂的非光滑学习问题（如鲁棒低秩矩阵恢复）的优化几何结构。