# Improving Matrix Exponential for Generative AI Flows: A Taylor-Based Approach Beyond Paterson--Stockmeyer

arXiv: https://arxiv.org/abs/2512.20777 · HTML: https://mlsanity.com/paper/2512.20777/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 43 references that are in the map.
- First posted on arXiv 23 Dec 2025.

## Builds on (most cited first, 11 of 11)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Simple and Controllable Music Generation](https://mlsanity.com/paper/2306.05284/) (2023)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Generalized Denoising Auto-Encoders as Generative Models](https://mlsanity.com/paper/1305.6663/) (2013)
- [Generative Models as a Data Source for Multiview Representation Learning](https://mlsanity.com/paper/2106.05258/) (2021)
- [Music ControlNet: Multiple Time-varying Controls for Music Generation](https://mlsanity.com/paper/2311.07069/) (2023)
- [Generative Flows with Matrix Exponential](https://mlsanity.com/paper/2007.09651/) (2020)

## Abstract (arXiv)

The matrix exponential is a fundamental operator in scientific computing and system simulation, with applications ranging from control theory and quantum mechanics to modern generative machine learning. While Pad'e approximants combined with scaling and squaring have long served as the standard, recent Taylor-based methods, which utilize polynomial evaluation schemes that surpass the classical Paterson--Stockmeyer technique, offer superior accuracy and reduced computational complexity. This paper presents an optimized Taylor-based algorithm for the matrix exponential, specifically designed for the high-throughput requirements of generative AI flows. We provide a rigorous error analysis and develop a dynamic selection strategy for the Taylor order and scaling factor to minimize computational effort under a prescribed error tolerance. Extensive numerical experiments demonstrate that our approach provides significant acceleration and maintains high numerical stability compared to existing state-of-the-art implementations. These results establish the proposed method as a highly efficient tool for large-scale generative modeling.