# Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances

arXiv: https://arxiv.org/abs/2503.02494 · HTML: https://mlsanity.com/paper/2503.02494/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 49 references that are in the map.
- First posted on arXiv 4 Mar 2025.

## Builds on (most cited first, 21 of 21)

- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [From Data to Decisions: Distributionally Robust Optimization is Optimal](https://mlsanity.com/paper/1704.04118/) (2017)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)
- [A Penalty-free Infeasible Approach for a Class of Nonsmooth Optimization Problems over the Stiefel Manifold](https://mlsanity.com/paper/2103.03514/) (2021)
- [A Manifold Proximal Linear Method for Sparse Spectral Clustering with Application to Single-Cell RNA Sequencing Data Analysis](https://mlsanity.com/paper/2007.09524/) (2020)
- [A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold](https://mlsanity.com/paper/2205.10500/) (2022)
- [Primal-Dual Optimization Algorithms over Riemannian Manifolds: an Iteration Complexity Analysis](https://mlsanity.com/paper/1710.02236/) (2017)
- [Manifold Proximal Point Algorithms for Dual Principal Component Pursuit and Orthogonal Dictionary Learning](https://mlsanity.com/paper/2005.02356/) (2020)
- [Wasserstein distributionally robust optimization and its tractable regularization formulations](https://mlsanity.com/paper/2402.03942/) (2024)
- [Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold](https://mlsanity.com/paper/2011.14781/) (2020)
- [A Riemannian Proximal Newton Method](https://mlsanity.com/paper/2304.04032/) (2023)
- [A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2401.11573/) (2024)
- [A Communication-Efficient and Privacy-Aware Distributed Algorithm for Sparse PCA](https://mlsanity.com/paper/2106.03320/) (2021)
- [Tight Error Bounds for the Sign-Constrained Stiefel Manifold](https://mlsanity.com/paper/2210.05164/) (2022)
- [Discrete Approximation of Two-Stage Stochastic and Distributionally Robust Linear Complementarity Problems](https://mlsanity.com/paper/1706.06847/) (2017)
- [Seeking Consensus on Subspaces in Federated Principal Component Analysis](https://mlsanity.com/paper/2012.03461/) (2020)

## Abstract (arXiv)

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.