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

Jorge Sastre, Daniel Faronbi, José Miguel Alonso, Peter Traver, Javier Ibáñez, Nuria Lloret

TL;DR

This paper targets the bottleneck of computing the matrix exponential in large-scale generative flows. It introduces a Taylor-based algorithm with dynamic selection of the Taylor degree and scaling parameter, underpinned by sharp error bounds for nonnormal matrices and advanced polynomial evaluation formulas that outperform Paterson-Stockmeyer. Through MATLAB benchmarks and PyTorch Glow experiments, it demonstrates substantial training speedups (3.9x–9.7x) and meaningful inference latency reductions, validating a portable, library-independent solution for high-throughput generative modeling. The approach combines theoretical error guarantees with practical, scalable implementations, making high-order Taylor approximations viable for real-world flow-based architectures.

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 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.

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

TL;DR

This paper targets the bottleneck of computing the matrix exponential in large-scale generative flows. It introduces a Taylor-based algorithm with dynamic selection of the Taylor degree and scaling parameter, underpinned by sharp error bounds for nonnormal matrices and advanced polynomial evaluation formulas that outperform Paterson-Stockmeyer. Through MATLAB benchmarks and PyTorch Glow experiments, it demonstrates substantial training speedups (3.9x–9.7x) and meaningful inference latency reductions, validating a portable, library-independent solution for high-throughput generative modeling. The approach combines theoretical error guarantees with practical, scalable implementations, making high-order Taylor approximations viable for real-world flow-based architectures.

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 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.
Paper Structure (17 sections, 3 theorems, 47 equations, 6 figures, 5 tables, 4 algorithms)

This paper contains 17 sections, 3 theorems, 47 equations, 6 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. The matrix exponential for generative flows
  3. Generative models
  4. Generative flows
  5. Proposed method
  6. Evaluation formulas of the Taylor-based matrix exponential approximations
  7. Error analysis
  8. Proposed algorithms
  9. Numerical results
  10. Numerical results in MATLAB
  11. Performance evaluation in the PyTorch generative flow environment
  12. Performance analysis in generative flow models
  13. Computational scaling and matrix dimensionality
  14. Training Performance and Scaling
  15. Inference and Sampling Performance
  16. ...and 2 more sections

Key Result

Theorem 1

Let $h_{l}(x)=\sum_{k \geq l} b_{k} x^{k}$ be a power series with radius of convergence $R$, and let $\tilde{h}_{l}(x)=\sum_{k \geq l} |b_{k}| x^{k}.$ For any matrix $A \in \mathbb{C}^{n \times n}$ with $\rho(A)<R$, if $a_{k}$ is an upper bound for $||A^{k}||$$(||A^{k}||\leq a_{k})$, $p \in \mathbb{ then

Figures (6)

  • Figure 1: Experimental results for matrices in MCT and EMP sets.
  • Figure 2: Experimental results for CIFAR10 dataset.
  • Figure 3: Experimental results for ImageNet32 dataset.
  • Figure 4: Experimental results for ImageNet64 dataset.
  • Figure 5: Diagram overview of flow model trained to generate images.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3