Optimization over the Stiefel manifold $\mathrm{St}(p,d)$, the set of $p \times d$ column-orthonormal matrices, is fundamental in statistics, machine learning, and scientific computing, yet remains challenging in the presence of non-convex, non-smooth, or black-box objectives. Existing methods largely rely on either convex relaxations or gradient-based Riemannian optimization, limiting applicability in derivative-free and highly multimodal settings. We propose \textsc{BOOOM} (Black-box Optimization Over Orthonormal Manifolds), a general-purpose framework for loss-function-agnostic optimization on $\mathrm{St}(p,d)$. The key idea is a global Givens rotation-based parametrization that maps the manifold to an unconstrained Euclidean angle space while preserving feasibility exactly. Building on this representation, BOOOM employs a structured, parallelizable, derivative-free search based on Recursive Modified Pattern Search, enabling systematic exploration through plane-wise rotations without requiring gradient information and facilitating escape from poor local optima. We establish a unified theoretical framework showing equivalence between angle-space and manifold optimization, transfer of stationarity, and global convergence in probability under mild conditions. Empirical results across diverse problems, including heterogeneous quadratic optimization, low-rank and sparse matrix decomposition, independent component analysis, and orthogonal joint diagonalization, among other widely studied settings, demonstrate strong performance relative to state-of-the-art methods, particularly in non-smooth and highly multimodal regimes. We further illustrate its practical utility through a novel supervised PCA formulation applied to metabolomics data in colorectal cancer.

翻译：在Steifel流形$\mathrm{St}(p,d)$（即$p \times d$列正交矩阵的集合）上的优化是统计学、机器学习与科学计算中的基础问题，但在面对非凸、非平滑或黑盒目标函数时仍具有挑战性。现有方法大多依赖于凸松弛或基于梯度的黎曼优化，这使得它们难以应用于无导数和高度多模态场景。我们提出\textsc{BOOOM}（正交流形上的黑盒优化），一个面向$\mathrm{St}(p,d)$上损失函数不可知优化的通用框架。其核心思想是基于全局Givens旋转的参数化方法，将流形映射到无约束的欧几里得角度空间，同时精确保持可行性。基于此表示，BOOOM采用基于递归修正模式搜索的、可并行化的结构化无导数搜索策略，通过平面旋转实现系统探索，无需梯度信息，并能有效摆脱劣质局部最优值。我们建立了一个统一的理论框架，证明角度空间优化与流形优化的等价性、平稳性的迁移性以及温和条件下的概率全局收敛性。在包括异质二次优化、低秩稀疏矩阵分解、独立成分分析和正交联合对角化等多种广泛研究的场景中，实验结果表明，与现有最优方法相比，BOOOM在非平滑和高度多模态问题中表现出显著优势。我们进一步通过应用于结直肠癌代谢组学数据的监督主成分分析新形式，展示了其实际应用价值。