# Positional Encoder Graph Quantile Neural Networks for Geographic Data

arXiv: https://arxiv.org/abs/2409.18865 · HTML: https://mlsanity.com/paper/2409.18865/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 38 references that are in the map.
- First posted on arXiv 27 Sep 2024.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [ETA Prediction with Graph Neural Networks in Google Maps](https://mlsanity.com/paper/2108.11482/) (2021)
- [Multi-Scale Representation Learning for Spatial Feature Distributions using Grid Cells](https://mlsanity.com/paper/2003.00824/) (2020)
- [Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets](https://mlsanity.com/paper/1406.7343/) (2014)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Calibrated and Sharp Uncertainties in Deep Learning via Density Estimation](https://mlsanity.com/paper/2112.07184/) (2021)
- [Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems](https://mlsanity.com/paper/1808.08798/) (2018)
- [Kriging Convolutional Networks](https://mlsanity.com/paper/2306.09463/) (2023)
- [Positional Encoder Graph Neural Networks for Geographic Data](https://mlsanity.com/paper/2111.10144/) (2021)
- [Autoregressive Quantile Flows for Predictive Uncertainty Estimation](https://mlsanity.com/paper/2112.04643/) (2021)
- [Basis-Function Models in Spatial Statistics](https://mlsanity.com/paper/2202.03660/) (2022)
- [Auxiliary-task learning for geographic data with autoregressive embeddings](https://mlsanity.com/paper/2006.10461/) (2020)
- [Expressive Monotonic Neural Networks](https://mlsanity.com/paper/2307.07512/) (2023)

## Built on by (most cited first, 1 of 1)

- [Quantile-Free Uncertainty Quantification in Graph Neural Networks](https://mlsanity.com/paper/2605.04847/) (2026)

## Abstract (arXiv)

Positional Encoder Graph Neural Networks (PE-GNNs) are among the most effective models for learning from continuous spatial data. However, their predictive distributions are often poorly calibrated, limiting their utility in applications that require reliable uncertainty quantification. We propose the Positional Encoder Graph Quantile Neural Network (PE-GQNN), a novel framework that combines PE-GNNs with Quantile Neural Networks, partially monotonic neural blocks, and post-hoc recalibration techniques. The PE-GQNN enables flexible and robust conditional density estimation with minimal assumptions about the target distribution, and it extends naturally to tasks beyond spatial data. Empirical results on benchmark datasets show that the PE-GQNN outperforms existing methods in both predictive accuracy and uncertainty quantification, without incurring additional computational cost. We also provide theoretical insights and identify important special cases arising from our formulation, including the PE-GNN.