【作者】 陈鹏；

【导师】 王学钦； 温灿红；

【作者基本信息】 中国科学技术大学 ， 统计学， 2025， 博士

【摘要】 在高维数据分析这一快速发展的领域中,稀疏优化已成为从复杂数据中提取结构模式与关键变量的核心工具。其基本思想是通过施加结构化约束（如l₀范数限制）在高维参数空间中实现变量选择一致性、估计效率与模型可解释性的平衡。然而,当模型固有约束与稀疏性约束之间存在复杂耦合时,优化问题的求解难度显著增加,从而对理论分析与计算实现均带来了重大挑战。本文围绕带有标准单纯形与稀疏性约束的非凸优化问题,系统探讨了其几何特性、算法设计与理论性质。首先,我们从单纯形的几何与统计结构出发,发现传统用于诱导稀疏性的Lasso惩罚项在单纯形可行域内退化为一个恒定的常数,从而失去其正则化作用。本文的分析表明,单纯形约束本身具有独特的“自正则化”性质,使得经验风险最小化解无需显式的惩罚项即可获得与Lasso估计量相当的统计表现。基于这一性质,我们对投影梯度（PG）下降方法在该约束下的收敛行为进行了理论刻画,证明了该方法能够以几何速度收敛到真实稀疏解的一个邻域,直至满足一定的统计精度,从而揭示了单纯形约束在高维统计学习中的内在稀疏诱导机制。然而,尽管投影迭代解具有较好的估计精度,但其通常是稠密的。为此,我们提出了PERMITS（Proj Ected g Radient Method w Ith Tail Screening）方法,通过在PG框架中嵌入尾部筛选机制,动态识别并舍弃可忽略的分量,从而在迭代过程中实现稀疏化。同时,我们将尾部筛选与信息准则相结合,使其能够自适应选择未知的稀疏度,该算法在提升计算效率的同时显著增强了变量选择的准确性。为了进一步突破高维计算的瓶颈,本文提出了剪接交换算法（Splicing Itera-tion Algorithm,SIA）。该方法仅需有限次全局梯度计算即可完成支撑集的迭代优化,大幅降低了对高维度的依赖。SIA从局部敏感性分析出发,利用原始–对偶信息度量候选支撑集的质量,并通过策略性的变量剪接与交换实现高效的支撑集更新,从而在保证估计精度的同时显著提升计算性能。在理论层面,本文系统建立了PERMITS与SIA的统计与计算性质,包括估计误差界、支撑集恢复性质及复杂度分析,结果表明这两种方法在高维环境下均具有良好的可扩展性与计算可行性。此外,通过模拟与实际数据实验,我们验证了本文所提出方法的理论预期与实际优越性,展示了其在高维稀疏优化及相关应用中的广泛潜力。更多还原

【Abstract】 In the rapidly evolving field of high-dimensional data analysis,sparse optimization has emerged as a fundamental tool for extracting structural patterns and identifying sig-nificant variables from complex datasets.Its core idea is to balance variable selection consistency,estimation efficiency,and model interpretability by imposing structured constraints,such as thel₀-norm constraint,within high-dimensional parameter spaces.However,the complex couple of internal constraint and sparsity constraint poses signif-icant challenges to the computational and theoretical developments of these methods.This paper systematically investigates the geometric characteristics,algorithmic design,and theoretical properties of nonconvex optimization problems with standard simplex and sparsity constraints.We begin by examining the geometric and statistical structure of the simplex set and find that the traditional Lasso penalty,which is commonly used to induce sparsity,reduces to a constant within the simplex,thereby losing its regularization effect.Our analysis further reveals that the simplex constraint itself exhibits an interesting self-regularizing property,enabling empirical risk minimizer to achieve Lasso-type statisti-cal performance without any explicit penalty.Building on this insight,we theoretically characterize the convergence behavior of the projected gradient（PG）method under sim-plex constraints ans show that it converges geometrically to a neighborhood of the true sparse solution,up to a certain level of statistical accuracy.This result highlights the intrinsic sparsity-inducing mechanism inherent in the simplex constraint within high-dimensional statistical learning.Although the PG iterates achieve satisfactory estimation accuracy,they are usually dense.To address this limitation,we propose PERMITS（Projected Gradient Method with Tail Screening）,which embeds a tail screening procedure into the PG framework to dynamically identify and discard negligible components during the iterations,thereby achieving natural sparsification.Combined with an information criterion,PERMITS can adaptively infer the unknown sparsity level,leading to substantial improvements in variable selection accuracy and computational efficiency.To further overcome the computational bottleneck in high-dimensional scenarios,we develop the Splicing Iteration Algorithm（SIA）,which only requires a finite num-ber of global gradient evaluations,thereby significantly alleviating dependence on the original high dimensionality.This algorithm adopts a local sensitivity analysis perspec-tive,providing an efficient analytical framework for understanding how small perturba-tions in optimization variables affect the objective function and constraint satisfaction.It leverages the primal-dual information to measure the qualities of candidate support sets,enabling iterative refinement of solutions through strategic variable selection and elimination.Our theoretical analysis provides a comprehensive characterization of both statis-tical and computational properties for the proposed PERMITS and SIA methods.We establish estimation error bounds and guarantee exact support set recovery,demon-strating that our algorithms can perfectly identify the true sparse structure under mild conditions.Additionally,our computational complexity analysis reveals favorable scal-ing properties,showing that our algorithms remain computationally tractable even in high-dimensional settings where traditional approaches may become intractable.To validate our theoretical findings and demonstrate the practical superiority of our meth-ods,we conduct extensive numerical experiments on both synthetic data,which enable controlled examination of algorithmic performance,and real-world benchmark datasets spanning multiple application domains,thereby providing evidence of their broad prac-tical applicability.更多还原