# A Gaussian Process Regression Model for Distribution Inputs

arXiv: https://arxiv.org/abs/1701.09055 · HTML: https://mlsanity.com/paper/1701.09055/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2017.

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

- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [Gaussian Processes on Distributions based on Regularized Optimal Transport](https://mlsanity.com/paper/2210.06574/) (2022)
- [Marginalising over Stationary Kernels with Bayesian Quadrature](https://mlsanity.com/paper/2106.07452/) (2021)
- [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024)
- [Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernels](https://mlsanity.com/paper/2212.01310/) (2022)
- [Learning System Parameters from Turing Patterns](https://mlsanity.com/paper/2108.08542/) (2021)
- [Distributional Gaussian Processes Layers for Out-of-Distribution Detection](https://mlsanity.com/paper/2206.13346/) (2022)
- [Improved learning theory for kernel distribution regression with two-stage sampling](https://mlsanity.com/paper/2308.14335/) (2023)
- [Adaptive Residual Transformation for Enhanced Feature-Based OOD Detection in SAR Imagery](https://mlsanity.com/paper/2411.00274/) (2024)
- [Distributional encoding for Gaussian process regression with qualitative inputs](https://mlsanity.com/paper/2506.04813/) (2025)
- [Computationally-efficient initialisation of GPs: The generalised variogram method](https://mlsanity.com/paper/2210.05394/) (2022)
- [Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty](https://mlsanity.com/paper/2603.17271/) (2026)

## Abstract (arXiv)

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling.