Riemannian Regression

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 ${\rho }_{ij}=1-S_{ij}$ and the induced subtraction $x_i\ominus x_j={\rho }_{ij}(x_i-x_j)\mathrm{.}$ A Riemannian center $g=x_{\lambda }$ is selected as a discrete Fréchet mean, and regression is performed on the Riemannian-centered variables $X_R=W{X}_{c,\lambda }$ and $y_R=W{y}_{c,\lambda }$, where $W=\mathrm{diag}({\rho }_{1\lambda },\dots ,{\rho }_{n\lambda })$. The resulting estimator has the weighted least-squares form ${\widehat{\beta }}_R=(X_{c,\lambda }^t{W}^2{X}_{c,\lambda }{)}^{-1}{X}_{c,\lambda }^t{W}^2{y}_{c,\lambda }\mathrm{.}$ 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.