用改进的泰勒方法加速生成式AI中的矩阵指数计算。
Improving Matrix Exponential for Generative AI Flows: A Taylor-Based Approach Beyond Paterson--Stockmeyer
- 基于泰勒展开，优化多项式计算策略超越经典方法。
- 动态选择阶数与缩放因子，在误差可控下提速显著。
- 适合大规模生成模型对高速矩阵运算的需求。
矩阵指数是科学计算与系统仿真中的基础算子，广泛应用于控制理论、量子力学及现代生成式机器学习。尽管佩德近似结合缩放与平方法长期作为标准，但近期基于泰勒的方法利用超越经典帕特森-斯托克迈尔技术的多项式求值方案，展现出更优精度与更低计算复杂度。本文提出一种专为生成式AI高吞吐需求设计的优化泰勒算法，提供严格的误差分析，并开发了动态选择泰勒阶数与缩放因子的策略，在给定误差容限下最小化计算开销。大量数值实验表明，该方法相比现有最先进实现具有显著加速效果且保持高数值稳定性。结果证明该方法是大规模生成建模中高效可靠的工具。
原文摘要 · Abstract (English)
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 Padé 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.
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