# Approximation capability of neural networks on spaces of probability measures and tree-structured domains

arXiv: https://arxiv.org/abs/1906.00764 · HTML: https://mlsanity.com/paper/1906.00764/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2019.

## Built on by (most cited first, 10 of 10)

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- [Measure-conditional Discriminator with Stationary Optimum for GANs and Statistical Distance Surrogates](https://mlsanity.com/paper/2101.06802/) (2021)
- [Explaining Classifiers Trained on Raw Hierarchical Multiple-Instance Data](https://mlsanity.com/paper/2208.02694/) (2022)
- [Malicious Internet Entity Detection Using Local Graph Inference](https://mlsanity.com/paper/2408.03287/) (2024)
- [ORIGAMI: A generative transformer architecture for predictions from semi-structured data](https://mlsanity.com/paper/2412.17348/) (2024)
- [DisKO: Deep Koopman Learning in Distribution Space from Unpaired Snapshots](https://mlsanity.com/paper/2609.34629/) (2026)

## Abstract (arXiv)

This paper extends the proof of density of neural networks in the space of continuous (or even measurable) functions on Euclidean spaces to functions on compact sets of probability measures. By doing so the work parallels a more then a decade old results on mean-map embedding of probability measures in reproducing kernel Hilbert spaces. The work has wide practical consequences for multi-instance learning, where it theoretically justifies some recently proposed constructions. The result is then extended to Cartesian products, yielding universal approximation theorem for tree-structured domains, which naturally occur in data-exchange formats like JSON, XML, YAML, AVRO, and ProtoBuffer. This has important practical implications, as it enables to automatically create an architecture of neural networks for processing structured data (AutoML paradigms), as demonstrated by an accompanied library for JSON format.