A dimensional R2 regression metric

arXiv cs.LG / 5/5/2026

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Key Points

The paper argues that the standard R² regression metric has major drawbacks, including a limit to two-dimensional inputs, loss of detailed accuracy structure by collapsing results into a single scalar, and sensitivity to low-variance noise that can produce large negative values.
It introduces “Dimensional R²” (Dim-R²), an extension designed to work with arbitrarily high-dimensional inputs and to present model accuracy in a multidimensional way.
Dim-R² is proposed to be less sensitive to noise channels, aiming to avoid misleading negative or hard-to-interpret scores.
The authors validate the approach on synthetic sinusoidal data and on three multidimensional regression datasets, showing improved interpretability and usefulness for guiding regression modeling.

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.