# EMPEROR: Efficient Moment-Preserving Representation of Distributions

arXiv: https://arxiv.org/abs/2509.16379 · HTML: https://mlsanity.com/paper/2509.16379/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 23 references that are in the map.
- First posted on arXiv 19 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Moment Matching for Multi-Source Domain Adaptation](https://mlsanity.com/paper/1812.01754/) (2018)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Fine-tuning CNN Image Retrieval with No Human Annotation](https://mlsanity.com/paper/1711.02512/) (2017)
- [Wasserstein Embedding for Graph Learning](https://mlsanity.com/paper/2006.09430/) (2020)
- [FSPool: Learning Set Representations with Featurewise Sort Pooling](https://mlsanity.com/paper/1906.02795/) (2019)
- [Covariance Pooling For Facial Expression Recognition](https://mlsanity.com/paper/1805.04855/) (2018)
- [Generalized Max Pooling](https://mlsanity.com/paper/1406.0312/) (2014)
- [Equivariant vs. Invariant Layers: A Comparison of Backbone and Pooling for Point Cloud Classification](https://mlsanity.com/paper/2306.05553/) (2023)

## Abstract (arXiv)

We introduce EMPEROR (Efficient Moment-Preserving Representation of Distributions), a mathematically rigorous and computationally efficient framework for representing high-dimensional probability measures arising in neural network representations. Unlike heuristic global pooling operations, EMPEROR encodes a feature distribution through its statistical moments. Our approach leverages the theory of sliced moments: features are projected onto multiple directions, lightweight univariate Gaussian mixture models (GMMs) are fit to each projection, and the resulting slice parameters are aggregated into a compact descriptor. We establish determinacy guarantees via Carleman's condition and the Cram'er-Wold theorem, ensuring that the GMM is uniquely determined by its sliced moments, and we derive finite-sample error bounds that scale optimally with the number of slices and samples. Empirically, EMPEROR captures richer distributional information than common pooling schemes across various data modalities, while remaining computationally efficient and broadly applicable.