Exact Invariance from Rank-Coordinate Scoring: Differentiable Ranking Without Permutation Relaxation
Edwin Nguthiru Ndiritu · Zenodo (CERN European Organization for Nuclear Research) · 2026
We present a method for training neural rankers that are exactly invariant to strictly monotone reparameterization of their input features. Each item is scored from the within-group ranks of its features rather than from their magnitudes, and these ranks are obtained through an exact sort whose gradient is carried by the values being sorted. Because a strictly monotone transformation preserves all within-group ranks, the induced ranking is unchanged. We prove this and verify it on three real datasets: under monotone feature transformations applied to a fixed test set, the Kendall correlation of the proposed ranker is identical to numerical precision, while a multilayer perceptron and two established differentiable sorters, NeuralSort and Sinkhorn, degrade by up to 0.58 in several cases below random ordering. The invariance arises from scoring in rank coordinates, not from the mechanism that makes the sort differentiable. The method trades a modest amount of clean-data accuracy for structural robustness to calibration shift, gain variation, and batch effects.