用沃瑟斯坦距离提升PCA的分布鲁棒性，解决不确定性下的主成分分析问题。
Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances
- 基于沃瑟斯坦距离构建分布鲁棒优化模型，将PCA转化为流形上的非光滑极小化问题。
- 提出平滑流形近端梯度算法，实现全局收敛，迭代复杂度为O(ε⁻³)。
- 实验验证方法有效且可扩展，证明分布鲁棒性对PCA的必要性和合理性。
我们研究了主成分分析（PCA）的分布鲁棒优化（DRO）模型，以应对潜在概率分布的不确定性。该模型导致一个非光滑的约束型极小极大优化问题，其中模糊集通过类型-2沃瑟斯坦距离刻画分布不确定性。我们证明内层最大化问题存在闭式解，从而将原问题等价重写为在Stiefel流形上的非光滑极小化问题，这一形式超出现有算法处理能力。为此，我们设计了一种高效的平滑流形近端梯度算法，理论分析表明其具有黎曼梯度一致性，并能全局收敛至非光滑问题的驻点。同时，我们给出了达到ε-近似驻点的迭代复杂度为O(ε⁻³)。最后，数值实验验证了算法的有效性与可扩展性，并强调采用DRO模型对PCA的必要性与合理性。
原文摘要 · Abstract (English)
We consider the distributionally robust optimization (DRO) model of principal component analysis (PCA) to account for uncertainty in the underlying probability distribution. The resulting formulation leads to a nonsmooth constrained min-max optimization problem, where the ambiguity set captures the distributional uncertainty by the type-$2$ Wasserstein distance. We prove that the inner maximization problem admits a closed-form optimal value. This explicit characterization equivalently reformulates the original DRO model into a minimization problem on the Stiefel manifold with intricate nonsmooth terms, a challenging formulation beyond the reach of existing algorithms. To address this issue, we devise an efficient smoothing manifold proximal gradient algorithm. Our analysis establishes Riemannian gradient consistency and global convergence of our algorithm to a stationary point of the nonsmooth minimization problem. We also provide the iteration complexity $O(ε^{-3})$ of our algorithm to achieve an $ε$-approximate stationary point. Finally, numerical experiments are conducted to validate the effectiveness and scalability of our algorithm, as well as to highlight the necessity and rationality of adopting the DRO model for PCA.
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