黎曼回归
Riemannian Regression
- University of Costa Rica（哥斯达黎加大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出黎曼回归框架，用数据依赖的局部加权差替代向量差，通过UMAP、ISOMAP或DBSCAN构造局部度量，保持线性形式并改变拟合几何，以减少局部异常影响并适应密度变化。
中文摘要 AI 辅助
经典线性回归假设预测变量空间的几何结构是欧几里得的，并且所有中心化观测值以相同的几何尺度对最小二乘拟合做出贡献。本文提出了黎曼回归（Riemannian Regression），这是一种回归框架，其中通常的向量差被由数据依赖的相似性结构诱导的局部加权差所替代。我们引入了一个广义框架，称为黎曼回归，将经典回归扩展到任何具有局部距离结构的数据。通过为数据表配备局部度量，我们调整回归模型以纳入流形几何。给定由UMAP、ISOMAP或DBSCAN获得的相似性矩阵$S=(S_{ij})$，我们定义相异性系数$\rho_{ij}=1-S_{ij}$以及诱导的减法$x_i\ominus x_j=\rho_{ij}(x_i-x_j)$。选择一个黎曼中心$g=x_\lambda$作为离散Fréchet均值，并在黎曼中心化变量$X_R=W X_{c,\lambda}$和$y_R=W y_{c,\lambda}$上进行回归，其中$W=\operatorname{diag}(\rho_{1\lambda},\ldots,\rho_{n\lambda})$。所得估计量具有加权最小二乘形式$\widehat\beta_R=(X_{c,\lambda}^{t}W^2X_{c,\lambda})^{-1}X_{c,\lambda}^{t}W^2y_{c,\lambda}$。所提出的方法保持了回归模型的线性形式，同时改变了拟合的几何结构。本文开发了三种构造局部度量的方法：基于UMAP的模糊相似性、基于ISOMAP的归一化测地距离以及基于DBSCAN的密度相似性。模拟示例和Abalone数据集说明了黎曼回归如何减少局部异常观测的影响并适应不同局部密度的区域。
英文摘要
Classical linear regression assumes that the relevant geometry of the predictor space is Euclidean and that all centered observations contribute to the least-squares fit in the same geometric scale. This paper proposes \emph{Riemannian Regression}, a regression framework in which the usual vector differences are replaced by locally weighted differences induced by a data-dependent similarity structure. We introduce a generalized framework, termed {\em Riemannian Regression}, extending classic regression to any data endowed with a local distance structure. By equipping data tables with local metrics, we adapt regression model to incorporate manifold geometry. Given a similarity matrix $S=(S_{ij})$, obtained from UMAP, ISOMAP, or DBSCAN \cite{mcinnes,isomap,dbscan}, we define the dissimilarity coefficient $ρ_{ij}=1-S_{ij}$ and the induced subtraction $ x_i\ominus x_j=ρ_{ij}(x_i-x_j). $ A Riemannian center $g=x_λ$ is selected as a discrete Fréchet mean, and regression is performed on the Riemannian-centered variables $X_R=W X_{c,λ}$ and $y_R=W y_{c,λ}$, where $W=\operatorname{diag}(ρ_{1λ},\ldots,ρ_{nλ})$. The resulting estimator has the weighted least-squares form $ \widehatβ_R=(X_{c,λ}^{t}W^2X_{c,λ})^{-1}X_{c,λ}^{t}W^2y_{c,λ}. $ The proposed approach preserves the linear form of the regression model while changing the geometry of the fit. The paper develops three ways to construct the local metric: UMAP-based fuzzy similarities, ISOMAP-based normalized geodesic distances, and DBSCAN-based density similarities. Simulated examples and the Abalone data set illustrate how Riemannian Regression can reduce the influence of locally anomalous observations and adapt to regions with different local densities.
