A dimensional R2 regression metric

2026-05-01

AI Summary

Traditional R² scoring is limited to low-dimensional inputs, struggles to reveal the structure of multidimensional prediction errors, and is highly sensitive to low-variance noise, often yielding uninterpretable negative values. This work proposes Dimensional R² (Dim-R²), a mathematically reformulated extension of R² that incorporates dimension-preserving normalization and variance stabilization mechanisms, enabling its application to arbitrary-dimensional tensor inputs. Dim-R² retains spatial structure information in predictive accuracy, substantially reduces sensitivity to noise, and provides an interpretable multidimensional performance profile. Experiments on synthetic sinusoidal signals and three real-world multidimensional regression datasets demonstrate that Dim-R² effectively captures spatial error patterns, thereby facilitating model diagnosis and optimization.

Abstract

R2 score is the standard metric for evaluating regression tasks, offering a normalized magnitude-agnostic measure of accuracy that captures variance. However, R2 has three key limitations: it is limited to at most two dimensional inputs, it reduces the score to a single scalar that hides rich patterns of prediction accuracy, and it is sensitive to low-variance noise channels which can yield large, uninterpretable negative values. We introduce the Dimensional R2 score (Dim-R2), a simple extension of R2 that accepts data of arbitrary dimensionality, provides a multidimensional view of accuracy, and reduces sensitivity to noise. We demonstrate its advantages on both synthetic sinusoidal data and three multidimensional regression datasets. Dim-R2 offers an interpretable and flexible metric that highlights patterns in regression accuracy, guiding regression modeling.

Problem

• R2 score
• regression evaluation
• dimensionality
• noise sensitivity
• accuracy patterns

Innovation

• Dimensional R2
• regression metric
• multidimensional evaluation
• noise robustness
• interpretability