Sparsity-constrained optimization underlies many problems in signal processing, statistics, and machine learning. State-of-the-art hard-thresholding (HT) algorithms rely on an appropriately selected continuous step-size parameter to ensure convergence. In this paper, we propose a naturally convergent iterative algorithm, SCOPE (Sparsity-Constrained Optimization via sPlicing itEration). The algorithm is capable of optimizing nonlinear differentiable objective functions that are strongly convex and smooth on low-dimensional subspaces. SCOPE replaces the gradient step with a splicing operation guided directly by the objective value, thereby eliminating the need to tune any continuous hyperparameter. Theoretically, it achieves a linear convergence rate and recovers the true support set when the sparsity level is correctly specified. We also establish parallel theoretical results without relying on restricted-isometry-property-type conditions. We apply SCOPE's versatility and power to solve sparse quadratic optimization, learn sparse classifiers, and recover sparse Markov networks for binary variables. With our C++ implementation of SCOPE, numerical experiments on these tasks show that it achieves superior support recovery performance, confirming both its algorithmic efficiency and theoretical guarantees.

翻译：稀疏约束优化是信号处理、统计学和机器学习中诸多问题的基础。现有最优的硬阈值（HT）算法依赖于适当选取的连续步长参数以确保收敛性。本文提出一种自然收敛的迭代算法SCOPE（基于拼接迭代的稀疏约束优化）。该算法能够优化在低维子空间上强凸且光滑的非线性可微目标函数。SCOPE用直接由目标值引导的拼接操作替代梯度步，从而消除对任何连续超参数调优的需求。理论上，该算法实现线性收敛速率，并在稀疏度正确指定时恢复真实支撑集。我们还在不依赖受限等距性质类条件的情况下建立了并行理论结果。我们应用SCOPE的通用性与能力求解稀疏二次优化、学习稀疏分类器以及恢复二值变量的稀疏马尔可夫网络。使用C++实现的SCOPE，在这些任务上的数值实验表明，它实现了优越的支撑恢复性能，同时验证了其算法效率与理论保证。