Hilbert Split: Geometry-Aware Data Splitting for Generalization Assessment

Data splitting is a commonly used technique for assessing how well predictive models generalize. The challenge is to preserve the distribution of the data relevant to prediction while limiting train–test similarities that can make evaluation overly optimistic. We introduce Hilbert Split, which identifies response-relevant predictors through sensitivity analysis. The local regression fits underlying this analysis provide a geometry for separating observations. A Hilbert ordering of the selected predictors and response organizes the data into local strata, within which this geometry guides test-point selection. This construction preserves distributional coverage while reducing train-test similarity. For randomized within-stratum selection, we establish sharper concentration of summary averages than simple random splitting and Wasserstein control of distributional discrepancy, with rates governed by the selected dimension.