# Deconditional Downscaling with Gaussian Processes

arXiv: https://arxiv.org/abs/2105.12909 · HTML: https://mlsanity.com/paper/2105.12909/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 52 references that are in the map.
- First posted on arXiv 27 May 2021.

## Builds on (most cited first, 18 of 18)

- [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)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [DeepSD: Generating High Resolution Climate Change Projections through Single Image Super-Resolution](https://mlsanity.com/paper/1703.03126/) (2017)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Learning from Aggregate Observations](https://mlsanity.com/paper/2004.06316/) (2020)
- [Convolutional conditional neural processes for local climate downscaling](https://mlsanity.com/paper/2101.07950/) (2021)
- [ClimAlign: Unsupervised statistical downscaling of climate variables via normalizing flows](https://mlsanity.com/paper/2008.04679/) (2020)
- [Multi-task Learning for Aggregated Data using Gaussian Processes](https://mlsanity.com/paper/1906.09412/) (2019)
- [Kernel Mean Shrinkage Estimators](https://mlsanity.com/paper/1405.5505/) (2014)
- [Variational Learning on Aggregate Outputs with Gaussian Processes](https://mlsanity.com/paper/1805.08463/) (2018)
- [Hyperparameter Learning via Distributional Transfer](https://mlsanity.com/paper/1810.06305/) (2018)
- [Spatially Aggregated Gaussian Processes with Multivariate Areal Outputs](https://mlsanity.com/paper/1907.08350/) (2019)
- [Multi-resolution Multi-task Gaussian Processes](https://mlsanity.com/paper/1906.08344/) (2019)
- [Inter-domain Deep Gaussian Processes](https://mlsanity.com/paper/2011.00415/) (2020)
- [Bayesian Deconditional Kernel Mean Embeddings](https://mlsanity.com/paper/1906.00199/) (2019)

## Built on by (most cited first, 24 of 24)

- [Looping in the Human Collaborative and Explainable Bayesian Optimization](https://mlsanity.com/paper/2310.17273/) (2023)
- [BayesIMP: Uncertainty Quantification for Causal Data Fusion](https://mlsanity.com/paper/2106.03477/) (2021)
- [Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Models](https://mlsanity.com/paper/2305.15167/) (2023)
- [Domain Generalisation via Imprecise Learning](https://mlsanity.com/paper/2404.04669/) (2024)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [An Overview of Causal Inference using Kernel Embeddings](https://mlsanity.com/paper/2410.22754/) (2024)
- [Exact Shapley Attributions in Quadratic-time for FANOVA Gaussian Processes](https://mlsanity.com/paper/2508.14499/) (2025)
- [Foundation Models for Environmental Science: A Survey of Emerging Frontiers](https://mlsanity.com/paper/2504.04280/) (2025)
- [Multi-Scale Graph Learning for Anti-Sparse Downscaling](https://mlsanity.com/paper/2505.01948/) (2025)
- [Interventional Processes for Causal Uncertainty Quantification](https://mlsanity.com/paper/2410.14483/) (2024)
- [All Models Are Miscalibrated, But Some Less So: Comparing Calibration with Conditional Mean Operators](https://mlsanity.com/paper/2502.11465/) (2025)
- [Sequential Decision Making on Unmatched Data using Bayesian Kernel Embeddings](https://mlsanity.com/paper/2210.13692/) (2022)
- [Turbine location-aware multi-decadal wind power predictions for Germany using CMIP6](https://mlsanity.com/paper/2408.14889/) (2024)
- [Indirect Query Bayesian Optimization with Integrated Feedback](https://mlsanity.com/paper/2412.13559/) (2024)
- [ActiveCQ: Active Estimation of Causal Quantities](https://mlsanity.com/paper/2509.24293/) (2025)
- [CLAM: Causal Spatial Disaggregation to Infer Local Effects From Coarse Data](https://mlsanity.com/paper/2608.08064/) (2026)
- [AODisaggregation: toward global aerosol vertical profiles](https://mlsanity.com/paper/2205.04296/) (2022)
- [Aggregated Multi-output Gaussian Processes with Knowledge Transfer Across Domains](https://mlsanity.com/paper/2206.12141/) (2022)
- [Bayesian Counterfactual Mean Embeddings and Off-Policy Evaluation](https://mlsanity.com/paper/2211.01518/) (2022)
- [Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledge](https://mlsanity.com/paper/2301.11214/) (2023)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [Anchor-MoE: A Mean-Anchored Mixture of Experts For Probabilistic Regression](https://mlsanity.com/paper/2508.16802/) (2025)
- [ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation](https://mlsanity.com/paper/2606.02247/) (2026)

## Abstract (arXiv)

Refining low-resolution (LR) spatial fields with high-resolution (HR) information, often known as statistical downscaling, is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an "inverse" of the conditional expectation, namely a deconditioning problem. In this work, we propose a Bayesian formulation of deconditioning which naturally recovers the initial reproducing kernel Hilbert space formulation from Hsu and Ramos (2019). We extend deconditioning to a downscaling setup and devise efficient conditional mean embedding estimator for multiresolution data. By treating conditional expectations as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.