The matrix exponential is a key operation in scientific computing and system simulation, with applications ranging from control theory and quantum mechanics to modern generative machine learning. Although Padé approximants combined with scaling and squaring have traditionally been the standard approach, recent advances in Taylor-based methods have shown that more efficient polynomial evaluation schemes can outperform the classical Paterson–Stockmeyer algorithm, both in accuracy and computational cost.
In this work, J. Sastre, D. Faronbi, J. M. Alonso, P. Traver, J. Ibáñez, and N. Lloret introduce an optimized Taylor-based algorithm for computing the matrix exponential, specifically developed to address the computational demands of generative AI flows. The method combines advanced matrix polynomial evaluation techniques with a rigorous error analysis and an adaptive strategy for selecting the Taylor approximation order and scaling factor. This allows the algorithm to adjust its computational effort to the properties of each input matrix while meeting a prescribed accuracy tolerance.
Beyond the numerical formulation itself, the main contribution lies in translating these algorithmic improvements into measurable gains in generative model training and inference. By reducing the number of matrix multiplications required without compromising numerical accuracy, the proposed approach addresses a significant computational bottleneck in flow-based architectures. Experiments on standard image datasets report training speedups ranging from 3.91× to 9.74× compared with the original implementation, while inference on batches of 128 samples is approximately 1.95× faster. These results illustrate how improvements in numerical algorithms can directly contribute to the efficiency and scalability of modern generative models.
Publication Details
- Title: Improving Matrix Exponential for Generative AI Flows: A Taylor-Based Approach Beyond Paterson–Stockmeyer
- Authors: J. Sastre, D. Faronbi, J. M. Alonso, P. Traver, J. Ibáñez, N. Lloret
- Preprint: arXiv:2512.20777 (v2, January 9 of 2026)
- Enlace: https://arxiv.org/abs/2512.20777
- Preprint (ResearchGate): https://www.researchgate.net/publication/399059963
Main Countributions
- Optimized Taylor algorithm for the matrix exponential, designed for generative AI flows.
- Error analysis and an adaptive strategy for selecting the Taylor approximation order and scaling factor.
- Significant speedups while maintaining high numerical stability compared with reference implementations.
Why This Matters
Generative AI flows rely on computationally intensive matrix operations. A faster and more stable method for computing the matrix exponential can reduce the computational cost of these models without compromising accuracy.