Data to 14 Oct 2026

Fast Differentiable Sorting and Ranking

arXiv 2002.08871 ↗First posted 20 Feb 2020stat.ML cs.LGcited by 197

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Abstract (arXiv)

The sorting operation is one of the most commonly used building blocks in computer programming. In machine learning, it is often used for robust statistics. However, seen as a function, it is piecewise linear and as a result includes many kinks where it is non-differentiable. More problematic is the related ranking operator, often used for order statistics and ranking metrics. It is a piecewise constant function, meaning that its derivatives are null or undefined. While numerous works have proposed differentiable proxies to sorting and ranking, they do not achieve the O(n log n) time complexity one would expect from sorting and ranking operations. In this paper, we propose the first differentiable sorting and ranking operators with O(n log n) time and O(n) space complexity. Our proposal in addition enjoys exact computation and differentiation. We achieve this feat by constructing differentiable operators as projections onto the permutahedron, the convex hull of permutations, and using a reduction to isotonic optimization. Empirically, we confirm that our approach is an order of magnitude faster than existing approaches and showcase two novel applications: differentiable Spearman's rank correlation coefficient and least trimmed squares.

Key facts. Cited by 197 papers in the map (data through 14 Oct 2026). First posted on arXiv 20 Feb 2020.

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Citing papers per year: 2021: 24; 2022: 26; 2023: 41; 2024: 43; 2025: 28; 2026: 35.

197 papers.

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2022Git Re-Basin: Merging Models modulo Permutation Symmetriescited by 471
2023Blind Image Quality Assessment via Vision-Language Correspondence: A Multitask Learning Perspectivecited by 280
2021Efficient and Modular Implicit Differentiationcited by 203
2023Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunitiescited by 130
2021Learning Optimal Conformal Classifierscited by 112
2022Equivariance with Learned Canonicalization Functionscited by 109
2021Blindly Assess Quality of In-the-Wild Videos via Quality-aware Pre-training and Motion Perceptioncited by 97
2023Uncertainty Quantification over Graph with Conformalized Graph Neural Networkscited by 79
2025VisualQuality-R1: Reasoning-Induced Image Quality Assessment via Reinforcement Learning to Rankcited by 79
2021Differentiable Patch Selection for Image Recognitioncited by 68
2021GrooMeD-NMS: Grouped Mathematically Differentiable NMS for Monocular 3D Object Detectioncited by 66
2022Training Uncertainty-Aware Classifiers with Conformalized Deep Learningcited by 64

38 more

2021Universal Approximation of Functions on Setscited by 62
2025Enhancing Time Series Forecasting through Selective Representation Spaces: A Patch Perspectivecited by 39
2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
2021Self-Supervised Online Reward Shaping in Sparse-Reward Environmentscited by 37
2022SIMPLE: A Gradient Estimator for k-Subset Samplingcited by 34
2023BayesDAG: Gradient-Based Posterior Inference for Causal Discoverycited by 33
2021Differentiable Sorting Networks for Scalable Sorting and Ranking Supervisioncited by 31
2025JarvisEvo: Towards a Self-Evolving Photo Editing Agent with Synergistic Editor-Evaluator Optimizationcited by 31
2021Meta-Calibration: Learning of Model Calibration Using Differentiable Expected Calibration Errorcited by 29
2022Backpropagation through Combinatorial Algorithms: Identity with Projection Workscited by 29
2022Learning with Combinatorial Optimization Layers: a Probabilistic Approachcited by 29
2023A Large-Scale Study of Probabilistic Calibration in Neural Network Regressioncited by 27
2024Provably Robust Conformal Prediction with Improved Efficiencycited by 27
2024Boosted Conformal Prediction Intervalscited by 27
2021ICON: Learning Regular Maps Through Inverse Consistencycited by 26
2023Fast, Differentiable and Sparse Top-k: a Convex Analysis Perspectivecited by 26
2024Distributional Preference Alignment of LLMs via Optimal Transportcited by 26
2022Monotonic Differentiable Sorting Networkscited by 25
2021Techniques for Symbol Grounding with SATNetcited by 24
2021Ranking the information content of distance measurescited by 22
2023Exploring the Naturalness of AI-Generated Imagescited by 22
2021Learning with Algorithmic Supervision via Continuous Relaxationscited by 21
2022Differentiable Top-k Classification Learningcited by 21
2024An Information Theoretic Perspective on Conformal Predictioncited by 21
2023ProtoGate: Prototype-based Neural Networks with Global-to-local Feature Selection for Tabular Biomedical Datacited by 18
2025FlexSelect: Flexible Token Selection for Efficient Long Video Understandingcited by 18
2021Implicit Generative Copulascited by 17
2022Flexible Differentiable Optimization via Model Transformationscited by 16
2025Improving Equivariant Networks with Probabilistic Symmetry Breakingcited by 16
2023BOtied: Multi-objective Bayesian optimization with tied multivariate rankscited by 15
2024Video Quality Assessment: A Comprehensive Surveycited by 15
2025Leveraging Allophony in Self-Supervised Speech Models for Atypical Pronunciation Assessmentcited by 15
2025Unified Uncertainty-Aware Diffusion for Multi-Agent Trajectory Modelingcited by 15
2021Self-Supervised Learning of Audio Representations from Permutations with Differentiable Rankingcited by 14
2023Differentiable Clustering with Perturbed Spanning Forestscited by 14
2026Learning to Evict from Key-Value Cachecited by 14
2022Optimizing generalized Gini indices for fairness in rankingscited by 13
2023Learning by Sorting: Self-supervised Learning with Group Ordering Constraintscited by 13

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