流形回归

中文摘要 AI 辅助

传统统计建模通常假设输入与输出变量之间存在一一映射关系，此处的“一一”指的是给定输入会产生唯一输出的预测意义。然而，许多科学与工程系统呈现非一一（noto）输入-输出关系：同一输入可能对应多个可接受的输出，因此传统统计模型可能不再适用。本文提出流形回归（manifold regression），一种针对此类非一一输入-输出关系的参数化建模框架，通过潜在流形表征底层输入-输出关系。该流形由未知有限维参数确定，我们采用带鲁棒求解器的正则化轮廓优化方法，从普通输入-输出观测中估计未知参数。流形学习完成后，通过沿指定坐标值对估计流形进行切片得到预测结果；由于切片可能包含多个潜在根，流形预测自然为集合值形式。该公式将预测学习扩展至传统一一统计模型之外，且包含经典回归与逆预测作为特例。本文建立理论结果以验证流形回归模型及其切片预测集的合理性，并通过案例研究展示其在代表性非一一场景中的稳定性能。

英文摘要

Conventional statistical modeling typically assumes a one-to-one mapping between input and output variables, where one-to-one is used in the predictive sense that a specified input results in a unique output. Many scientific and engineering systems, however, exhibit non-one-to-one(noto) input-output relations. In a noto relation, the same input may correspond to multiple admissible outputs, so the conventional statistical models may not be appropriate. This paper develops manifold regression, a parametric modeling framework for such noto input-output relations, which represents the underlying input-output relation through a latent manifold. The manifold is specified upto an unknown finite dimensional parameter and we estimate the unknown parameters from ordinary input-output observations by regularized profile optimization with a robust solver. Once the manifold has been learned, prediction is obtained by slicing the estimated manifold along a specified coordinate value. Because a slice may contain multiple latent roots, the resulting manifold prediction is naturally set-valued. This formulation extends predictive learning beyond conventional one-to-one statistical models while containing classical regression and inverse prediction as special cases. Theoretical results are established to justify the manifold regression model and its sliced prediction sets; and case studies are used to demonstrate stable performance across representative noto cases.

发表机构

• Department of Statistics, Purdue University（普渡大学统计系）

机构由 AI 辅助整理，请以论文原文为准。