基于投影重采样的多元非线性降维

On a Projective Resampling Method for Nonlinear Dimension Reduction with Multivariate Responses

摘要: 本文研究响应变量和预测因子均为向量的充分降维问题。投影重采样方法的核心思想是将多元响应投影到随机采样的方向上，以获取标量响应的样本，并反复应用单变量响应的降维方法来解决问题。该方法已被证明对多元线性降维有效。本文将投影重采样方法推广到非线性情境，并通过核映射提出了四种新的估计方法。研究结果表明，新方法具有优良的性质，并能在温和条件下完整恢复降维空间。最后，通过数值模拟和真实数据集分析验证了所提方法的有效性和可行性。

Abstract: This paper addresses the problem of sufficient dimension reduction where both the response and predictor are vectors. The core idea of projective resampling method is to project the multivariate responses along randomly sampled directions to obtain samples of scalar-valued responses. A univariate-response dimension reduction method is then applied repeatedly for solving the problem. This has proven effective for multivariate linear dimension reduction. In this paper, we extend the projective resampling method to nonlinear scenarios and use the mapping induced by kernels to develop four novel estimation methods. The research results suggest that the new methods exhibit excellent properties and ensure full recovery of the dimension reduction space under mild conditions. Finally, we validate the effectiveness and feasibility of the proposed methods through numerical simulations and real data analysis.