# Structural Incompatibility of Differentiable Sorting and Within-Vector Rank Normalization

arXiv: https://arxiv.org/abs/2512.22587 · HTML: https://mlsanity.com/paper/2512.22587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 17 references that are in the map.
- First posted on arXiv 27 Dec 2025.

## Builds on (most cited first, 11 of 11)

- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [Differentiable Sorting Networks for Scalable Sorting and Ranking Supervision](https://mlsanity.com/paper/2105.04019/) (2021)
- [Monotonic Differentiable Sorting Networks](https://mlsanity.com/paper/2203.09630/) (2022)
- [LapSum -- One Method to Differentiate Them All: Ranking, Sorting and Top-k Selection](https://mlsanity.com/paper/2503.06242/) (2025)
- [Generalized Neural Sorting Networks with Error-Free Differentiable Swap Functions](https://mlsanity.com/paper/2310.07174/) (2023)

## Abstract (arXiv)

We show that differentiable sorting and ranking operators are structurally incompatible with within-vector rank normalization. We formalize admissibility through monotone invariance (C1), batch independence (C2), and a rank-space stability condition (C3). Gap-sensitive relaxations such as SoftSort violate (C1) by a quantitative margin that depends on the temperature and input scale. Batchwise rank relaxations such as SinkhornSort violate (C2): the same sample can be assigned outputs arbitrarily close to 0 or 1 depending solely on batch context. Condition (C3) implies (C1) under the rank representation used here and should not be read as a third independent failure mode. We also characterize the admissible class: any admissible operator must factor through the rank representation via a Lipschitz function.