In this work, we propose an optimization framework for estimating a sparse robust one-dimensional subspace. Our objective is to minimize both the representation error and the penalty, in terms of the l1-norm criterion. Given that the problem is NP-hard, we introduce a linear relaxation-based approach. Additionally, we present a novel fitting procedure, utilizing simple ratios and sorting techniques. The proposed algorithm demonstrates a worst-case time complexity of $O(n^2 m \log n)$ and, in certain instances, achieves global optimality for the sparse robust subspace, thereby exhibiting polynomial time efficiency. Compared to extant methodologies, the proposed algorithm finds the subspace with the lowest discordance, offering a smoother trade-off between sparsity and fit. Its architecture affords scalability, evidenced by a 16-fold improvement in computational speeds for matrices of 2000x2000 over CPU version. Furthermore, this method is distinguished by several advantages, including its independence from initialization and deterministic and replicable procedures. Furthermore, this method is distinguished by several advantages, including its independence from initialization and deterministic and replicable procedures. The real-world example demonstrates the effectiveness of algorithm in achieving meaningful sparsity, underscoring its precise and useful application across various domains.

翻译：本文提出了一种用于估计稀疏鲁棒一维子空间的优化框架。我们的目标是在l1范数准则下，同时最小化表示误差与惩罚项。鉴于该问题为NP难问题，我们引入了基于线性松弛的方法。此外，我们提出了一种新颖的拟合流程，利用简单的比值与排序技术。所提算法在最坏情况下的时间复杂度为$O(n^2 m \log n)$，并且在某些情形下能够为稀疏鲁棒子空间实现全局最优，从而展现出多项式时间效率。与现有方法相比，所提算法能找到不一致性最低的子空间，在稀疏性与拟合度之间提供更平滑的权衡。其架构具备可扩展性，在2000×2000矩阵上的计算速度相比于CPU版本提升了16倍。此外，该方法具有若干优势，包括不依赖初始值、流程确定且可复现。实际案例展示了算法在实现有意义的稀疏性方面的有效性，凸显了其在多个领域的精确与实用应用。