# Generative Quantile Regression with Variability Penalty

arXiv: https://arxiv.org/abs/2301.03661 · HTML: https://mlsanity.com/paper/2301.03661/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2023.

## Built on by (most cited first, 4 of 4)

- [Deep Generative Quantile Bayes](https://mlsanity.com/paper/2410.08378/) (2024)
- [Nonparametric estimation of conditional probability distributions using a generative approach based on conditional push-forward neural networks](https://mlsanity.com/paper/2511.14455/) (2025)
- [Neural-g: A Deep Learning Framework for Mixing Density Estimation](https://mlsanity.com/paper/2406.05986/) (2024)
- [A Monotone Single-Index Modal Regression Powered by Deep Neural Networks for Non-Gaussian Periodontal Data](https://mlsanity.com/paper/2505.02153/) (2025)

## Abstract (arXiv)

Quantile regression and conditional density estimation can reveal structure that is missed by mean regression, such as multimodality and skewness. In this paper, we introduce a deep learning generative model for joint quantile estimation called Penalized Generative Quantile Regression (PGQR). Our approach simultaneously generates samples from many random quantile levels, allowing us to infer the conditional distribution of a response variable given a set of covariates. Our method employs a novel variability penalty to avoid the problem of vanishing variability, or memorization, in deep generative models. Further, we introduce a new family of partial monotonic neural networks (PMNN) to circumvent the problem of crossing quantile curves. A major benefit of PGQR is that it can be fit using a single optimization, thus bypassing the need to repeatedly train the model at multiple quantile levels or use computationally expensive cross-validation to tune the penalty parameter. We illustrate the efficacy of PGQR through extensive simulation studies and analysis of real datasets. Code to implement our method is available at https://github.com/shijiew97/PGQR.