One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators

2026-05-11 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Data Science & Analytics · Depth: Expert, extended

Summary

This paper introduces a novel operator learning framework for probabilistic conditioning, defining two continuous operators: the kernel conditioning operator (mapping a joint density to its conditional kernel) and the in-context conditioning operator (mapping a joint density and a query variable to the conditional density). The authors prove that these operators can be universally approximated by neural operators, specifically Fourier Neural Operators (FNOs) and Transformer Neural Operators (TNOs), to arbitrary accuracy over compact sets of continuous densities, including Hölder densities and Gaussian mixtures. This framework enables amortization across joint distributions, eliminating the need for retraining for each new input density. Numerical experiments demonstrate that trained neural operators can accurately implement conditioning for Gaussian mixture test problems, outperforming traditional plug-in estimators in certain settings, particularly for correlated Gaussian (K=1) KDEs.

Key takeaway

For AI Scientists and Machine Learning Engineers developing probabilistic models, this work suggests a shift towards operator learning for conditional inference. Your teams should explore integrating neural operators, such as FNOs or TNOs, to create "foundation models for conditioning." This approach allows for amortized inference, significantly reducing computational overhead by avoiding retraining for each new joint distribution, thereby accelerating tasks like Bayesian inference and data assimilation.

Key insights

Neural operators can universally approximate probabilistic conditioning, enabling amortized inference across diverse joint distributions.

Principles

• Conditioning operators are continuous over suitable density classes.
• Continuity enables universal approximation by neural operators.
• Amortization avoids retraining for new input densities.

Method

The method formalizes conditioning as a nonlinear operator, proves its continuity, and then applies universal approximation theorems for neural operators (FNOs, TNOs) to learn this operator across various density inputs.

In practice

• Use neural operators for rapid conditional inference on unseen joint densities.
• Apply to sequential Bayesian inference tasks like nonlinear filtering.
• Consider for optimal experimental design problems.

Topics

• Neural Operators
• Probabilistic Conditioning
• Universal Approximation Theorem
• Operator Learning Framework
• Gaussian Mixture Models

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.