Sliced Inverse Regression in Metric Spaces

: In this article, we propose a general nonlinear sufficient dimension reduction (SDR) framework when both the predictor and response lie in some general metric spaces. We construct reproducing kernel Hilbert spaces whose kernels are fully determined by the distance functions of the metric spaces, then leverage the inherent structures of these spaces to define a nonlinear SDR framework. We adapt the classical sliced inverse regression of Li (1991) within this framework for the metric space data. We build the estimator based on the corresponding linear operators, and show it recovers the regression information unbiasedly. We derive the estimator at both the operator level and under a coordinate system, and also establish its convergence rate. We illustrate the proposed method with both synthetic and real datasets exhibiting non-Euclidean geometry.

在本文中，当预测变量和响应变量都位于某些一般度量空间时，我们提出了一个通用的非线性充分降维（SDR）框架。我们构建了再生核希尔伯特空间，其核完全由度量空间的距离函数确定，然后利用这些空间的固有结构来定义一个非线性SDR框架。我们在这个框架内对度量空间数据采用了李（1991）的经典切片逆回归方法。我们基于相应的线性算子构建估计量，并表明它无偏地恢复回归信息。我们在算子层面和坐标系下都推导出了估计量，并且还确定了其收敛速度。我们用呈现非欧几里得几何的合成数据集和真实数据集来说明所提出的方法。