Nonsmooth composite optimization with orthogonality constraints has a wide range of applications in statistical learning and data science. However, this problem is challenging due to its nonsmooth objective and computationally expensive, non-convex constraints. In this paper, we propose a new approach called \textbf{OBCD}, which leverages Block Coordinate Descent to address these challenges. \textbf{OBCD} is a feasible method with a small computational footprint. In each iteration, it updates $k$ rows of the solution matrix, where $k \geq 2$, by globally solving a small nonsmooth optimization problem under orthogonality constraints. We prove that the limiting points of \textbf{OBCD}, referred to as (global) block-$k$ stationary points, offer stronger optimality than standard critical points. Furthermore, we show that \textbf{OBCD} converges to $\epsilon$-block-$k$ stationary points with an ergodic convergence rate of $\mathcal{O}(1/\epsilon)$. Additionally, under the Kurdyka-Lojasiewicz (KL) inequality, we establish the non-ergodic convergence rate of \textbf{OBCD}. We also extend \textbf{OBCD} by incorporating breakpoint searching methods for subproblem solving and greedy strategies for working set selection. Comprehensive experiments demonstrate the superior performance of our approach across various tasks.

翻译：正交约束下的非光滑复合优化在统计学习和数据科学中具有广泛应用。然而，由于目标函数的非光滑性以及计算代价高昂的非凸约束，该问题具有较大挑战性。本文提出一种名为 \textbf{OBCD} 的新方法，该方法利用块坐标下降法应对这些挑战。\textbf{OBCD} 是一种计算开销较小的可行方法。在每次迭代中，它通过全局求解一个正交约束下的小规模非光滑优化问题，更新解矩阵的 $k$ 行（其中 $k \geq 2$）。我们证明 \textbf{OBCD} 的极限点（称为（全局）块-$k$ 稳定点）比标准临界点具有更强的优化性。此外，我们证明 \textbf{OBCD} 以 $\mathcal{O}(1/\epsilon)$ 的遍历收敛速率收敛到 $\epsilon$-块-$k$ 稳定点。同时，在 Kurdyka-Lojasiewicz (KL) 不等式条件下，我们建立了 \textbf{OBCD} 的非遍历收敛速率。我们还通过引入断点搜索方法用于子问题求解，以及贪心策略用于工作集选择，对 \textbf{OBCD} 进行了扩展。综合实验表明，我们的方法在多种任务上均表现出优越性能。