提出新优化方法，在流形约束下高效求解光滑问题。
Manifold constrained steepest descent for smooth and closed-set optimization
- 在流形上直接构造下降方向，避免反复求解子问题。
- 理论保证收敛速度达O(log T / √T)，适用于闭集约束。
- 适合需要流形优化的机器学习与信号处理任务。
我们研究在具有光滑嵌入流形结构的可行集上最小化光滑函数的问题，使用线性最小化算子（LMO）在用户指定范数下确定搜索方向。然而，将LMO限制在切空间时可能需要迭代内层求解。本文提出曼德尔约束最速下降法（MCSD）及其切空间投影变体MCSD-TP，避免了切空间LMO子问题的迭代求解。在Stiefel流形上的谱范数特例得到SPEL方法，可通过矩阵符号计算高效实现。在标准正则性假设下，两种方法均达到O(log T / √T)的最优迭代点平稳性界。对于闭可行集，混合方法MCSD–PGD结合平滑方法与投影梯度下降；在相应局部光滑条件下，所有聚点均为Bouligand平稳点。实验在Stiefel约束主成分分析、加权低秩逼近和稀疏相位恢复中验证了所提方法的有效性。
原文摘要 · Abstract (English)
We study minimization of smooth functions over feasible sets that have smooth embedded-manifold structure throughout or only on selected regions, using linear minimization oracles (LMOs) to determine search directions under user-chosen norms. Restricting an LMO to a tangent space, however, can require an iterative inner solve. We propose \emph{Manifold Constrained Steepest Descent} (MCSD) and a tangent-projected variant, MCSD-TP, which avoid solving tangent-space LMO subproblems iteratively. The spectral-norm specialization of MCSD on the Stiefel manifold yields \emph{SPEL}, which admits an efficient implementation using matrix-sign computations. Under standard regularity assumptions, both methods attain an \(O(\log T/\sqrt T)\) best-iterate stationarity bound. For closed feasible sets, the hybrid MCSD--PGD combines either smooth method with projected gradient descent; under the corresponding local smoothness condition, every accumulation point is Bouligand stationary. Experiments on Stiefel-constrained PCA, weighted low-rank approximation, and sparse phase retrieval illustrate the proposed methods.
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