Beyond Contraction: Geometry-Faithful Supervised Dimensionality Reduction for Data Visualization

Supervised dimensionality reduction (DR) is widely used in visualization to reveal task-specific structure in high-dimensional data. However, in regression settings with continuous supervision, existing methods often improve signal readability at the cost of severe neighborhood distortion, limiting the reliability of the resulting visualization. To understand this trade-off, we provide a theoretical analysis that characterizes the relationship between geometric faithfulness and signal readability through two finite-sample proxies for the readout's bi-Lipschitz properties: contraction (co-Lipschitz) and smoothness (Lipschitz). We show that contraction can impose a sharp geometric distortion floor in many existing methods, whereas smoothness alone is sufficient to support effective visualization. In particular, directly regularizing smoothness yields first-order regularity gains with only second-order geometry loss, under geometry-first initialization near a local geometry optimum. Guided by this insight, we propose \sys (\acroinit{I}nterpretable \acroinit{S}upervised \acroinit{DR}), a nonparametric method that preserves geometric structure while enforcing readout smoothness. Experiments on HPDv3 and Tabula Muris show that \sys improves the geometry--signal Pareto frontier, reducing local signal variation while retaining faithful neighborhoods compared with representative supervised DR baselines. Our code is available at \url{https://anonymous.4open.science/r/ISDR-preview}.