基于凸对偶的单调曲线估计
Monotone Curve Estimation via Convex Duality
AI总结：
该研究基于最优传输理论提出带理论保证的单调主曲线框架，采用早停策略结合神经网络实现，模拟与真实数据实验验证其在单调结构数据上的准确性与鲁棒性，为相关应用提供新方法。
摘要：
主曲线是通过一维光滑连续表示揭示数据潜在结构的有力工具。本文基于最优传输理论，提出了一种受单调性约束的新型主曲线框架，并给出了严格的理论证明。我们为所提出的单调曲线估计建立了统计保证，包括期望经验均方误差和广义均方误差，同时证明了此类估计的存在性。这些统计基础为采用机器学习中流行的早停策略，结合神经网络实现我们的数值算法提供了依据。全面的模拟研究表明，当数据呈现单调结构时，所提出的单调曲线估计在准确性上优于现有对比方法。此外，通过两个实际应用——铜、金、银的未来价格预测，以及牛油果价格与销量分析——我们强调了该曲线估计对变量变换的鲁棒性，进一步证实了其在含噪声复杂数据集上的有效适用性。我们认为，这一单调曲线拟合框架在存在内在单调关系或需要施加单调约束的众多应用中具有巨大潜力。
英文摘要：
A principal curve serves as a powerful tool for uncovering underlying structures of data through 1-dimensional smooth and continuous representations. On the basis of optimal transport theories, this paper introduces a novel principal curve framework constrained by monotonicity with rigorous theoretical justifications. We establish statistical guarantees for our monotone curve estimate, including expected empirical and generalized mean squared errors, while proving the existence of such estimates. These statistical foundations justify adopting the popular early stopping procedure in machine learning to implement our numeric algorithm with neural networks. Comprehensive simulation studies reveal that the proposed monotone curve estimate outperforms competing methods in terms of accuracy when the data exhibits a monotonic structure. Moreover, through two real-world applications on future prices of copper, gold, and silver, and avocado prices and sales volume, we underline the robustness of our curve estimate against variable transformation, further confirming its effective applicability for noisy and complex data sets. We believe that this monotone curve-fitting framework offers significant potential for numerous applications where monotonic relationships are intrinsic or need to be imposed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
