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

TL;DR

This paper introduces an optimized Taylor-based algorithm for computing the matrix exponential, tailored for generative AI flows, offering improved accuracy, efficiency, and stability over traditional methods like Padé approximants.

Contribution

The paper develops a novel Taylor-based matrix exponential method with dynamic parameter selection, outperforming classical techniques in speed and stability for large-scale generative models.

Findings

Significant acceleration over existing methods.

High numerical stability in large-scale applications.

Effective dynamic parameter selection strategy.

Abstract

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.