# Deep Non-Crossing Quantiles through the Partial Derivative

arXiv: https://arxiv.org/abs/2201.12848 · HTML: https://mlsanity.com/paper/2201.12848/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 30 Jan 2022.

## Builds on (most cited first, 13 of 13)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Distributional Reinforcement Learning with Quantile Regression](https://mlsanity.com/paper/1710.10044/) (2017)
- [Implicit Quantile Networks for Distributional Reinforcement Learning](https://mlsanity.com/paper/1806.06923/) (2018)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Fully Parameterized Quantile Function for Distributional Reinforcement Learning](https://mlsanity.com/paper/1911.02140/) (2019)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Monotonic Calibrated Interpolated Look-Up Tables](https://mlsanity.com/paper/1505.06378/) (2015)
- [Modelling heterogeneous distributions with an Uncountable Mixture of Asymmetric Laplacians](https://mlsanity.com/paper/1910.12288/) (2019)
- [Single-Model Uncertainties for Deep Learning](https://mlsanity.com/paper/1811.00908/) (2018)
- [Monotonic Cardinality Estimation of Similarity Selection: A Deep Learning Approach](https://mlsanity.com/paper/2002.06442/) (2020)

## Built on by (most cited first, 12 of 12)

- [Fast Nonlinear Vector Quantile Regression](https://mlsanity.com/paper/2205.14977/) (2022)
- [Conformal Thresholded Intervals for Efficient Regression](https://mlsanity.com/paper/2407.14495/) (2024)
- [Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise](https://mlsanity.com/paper/2406.03258/) (2024)
- [Retrospective Uncertainties for Deep Models using Vine Copulas](https://mlsanity.com/paper/2302.12606/) (2023)
- [Nonparametric Quantile Regression: Non-Crossing Constraints and Conformal Prediction](https://mlsanity.com/paper/2210.10161/) (2022)
- [Estimation of Non-Crossing Quantile Regression Process with Deep ReQU Neural Networks](https://mlsanity.com/paper/2207.10442/) (2022)
- [Adaptive Conditional Quantile Neural Processes](https://mlsanity.com/paper/2305.18777/) (2023)
- [Censored Quantile Regression Neural Networks for Distribution-Free Survival Analysis](https://mlsanity.com/paper/2205.13496/) (2022)
- [OrderFusion: Encoding Orderbook for End-to-End Probabilistic Intraday Electricity Price Forecasting](https://mlsanity.com/paper/2502.06830/) (2025)
- [Neural Optimal Transport Meets Multivariate Conformal Prediction](https://mlsanity.com/paper/2509.25444/) (2025)
- [fastkqr: A Fast Algorithm for Kernel Quantile Regression](https://mlsanity.com/paper/2408.05393/) (2024)
- [Multi-Quantile Regression for Extreme Precipitation Downscaling](https://mlsanity.com/paper/2605.12762/) (2026)

## Abstract (arXiv)

Quantile Regression (QR) provides a way to approximate a single conditional quantile. To have a more informative description of the conditional distribution, QR can be merged with deep learning techniques to simultaneously estimate multiple quantiles. However, the minimisation of the QR-loss function does not guarantee non-crossing quantiles, which affects the validity of such predictions and introduces a critical issue in certain scenarios. In this article, we propose a generic deep learning algorithm for predicting an arbitrary number of quantiles that ensures the quantile monotonicity constraint up to the machine precision and maintains its modelling performance with respect to alternative models. The presented method is evaluated over several real-world datasets obtaining state-of-the-art results as well as showing that it scales to large-size data sets.