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

• URL: http://arxiv.org/abs/2512.20777v1
• Date: Tue, 23 Dec 2025 21:25:40 GMT
• Title: Improving Matrix Exponential for Generative AI Flows: A Taylor-Based Approach Beyond Paterson--Stockmeyer
• Abstract summary: The matrix exponential is a fundamental operator in scientific computing and system simulation.<n>This paper presents a Taylor-based algorithm for the matrix exponential, specifically designed for the high- throughput requirements of generative AI flows.
• Score: 1.228544477050145
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Machine Learning-Assisted High-Dimensional Matrix Estimation [19.738577017139345]
  We present a solution procedure based on the Linearized Alternating Direction Method of Multipliers (LADMM)<n>We then introduce learnable parameters and model the proximal operators in the iterative scheme with neural networks, thereby improving estimation accuracy and accelerating convergence.
  arXiv  Detail & Related papers  (2026-03-30T12:15:59Z)
• Natural Hypergradient Descent: Algorithm Design, Convergence Analysis, and Parallel Implementation [5.754044493040163]
  Natural Hypergradient Descent (NHGD) is a new method for solving bilevel optimization problems.<n>Our main theoretical contribution establishes high-probability error bounds and sample complexity guarantees for NHGD.<n> Empirical evaluations on representative bilevel learning tasks demonstrate the practical advantages of NHGD.
  arXiv  Detail & Related papers  (2026-02-11T14:31:33Z)
• ODELoRA: Training Low-Rank Adaptation by Solving Ordinary Differential Equations [54.886931928255564]
  Low-rank adaptation (LoRA) has emerged as a widely adopted parameter-efficient fine-tuning method in deep transfer learning.<n>We propose a novel continuous-time optimization dynamic for LoRA factor matrices in the form of an ordinary differential equation (ODE)<n>We show that ODELoRA achieves stable feature learning, a property that is crucial for training deep neural networks at different scales of problem dimensionality.
  arXiv  Detail & Related papers  (2026-02-07T10:19:36Z)
• Online Inference of Constrained Optimization: Primal-Dual Optimality and Sequential Quadratic Programming [55.848340925419286]
  We study online statistical inference for the solutions of quadratic optimization problems with equality and inequality constraints.<n>We develop a sequential programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing an approximation of the objective and a linear approximation of the constraints.<n>We show that our method global almost moving-average convergence and exhibits local normality with an optimal primal-dual limiting matrix in the sense of Hjek and Le Cam.
  arXiv  Detail & Related papers  (2025-11-27T06:16:17Z)
• A Robust and Non-Iterative Tensor Decomposition Method with Automatic Thresholding [1.9336815376402718]
  This study proposes a novel low-rank approximation method that eliminates both prior rank specification and iterative optimization.<n>The method applies statistical singular value hard thresholding to each mode-wise unfolded matrix to automatically extract statistically significant components.<n> Simulation experiments demonstrate that the proposed method outperforms conventional approaches.
  arXiv  Detail & Related papers  (2025-05-09T17:30:16Z)
• Adaptive Error-Bounded Hierarchical Matrices for Efficient Neural Network Compression [0.0]
  This paper introduces a dynamic, error-bounded hierarchical matrix (H-matrix) compression method tailored for Physics-Informed Neural Networks (PINNs) The proposed approach reduces the computational complexity and memory demands of large-scale physics-based models while preserving the essential properties of the Neural Tangent Kernel (NTK) Empirical results demonstrate that this technique outperforms traditional compression methods, such as Singular Value Decomposition (SVD), pruning, and quantization, by maintaining high accuracy and improving generalization capabilities.
  arXiv  Detail & Related papers  (2024-09-11T05:55:51Z)
• Operator SVD with Neural Networks via Nested Low-Rank Approximation [19.562492156734653]
  This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition. New techniques called emphnesting for learning the top-$L$ singular values and singular functions in the correct order. We demonstrate the effectiveness of the proposed framework for use cases in computational physics and machine learning.
  arXiv  Detail & Related papers  (2024-02-06T03:06:06Z)
• Machine Learning Insides OptVerse AI Solver: Design Principles and Applications [74.67495900436728]
  We present a comprehensive study on the integration of machine learning (ML) techniques into Huawei Cloud's OptVerse AI solver. We showcase our methods for generating complex SAT and MILP instances utilizing generative models that mirror multifaceted structures of real-world problem. We detail the incorporation of state-of-the-art parameter tuning algorithms which markedly elevate solver performance.
  arXiv  Detail & Related papers  (2024-01-11T15:02:15Z)
• GloptiNets: Scalable Non-Convex Optimization with Certificates [61.50835040805378]
  We present a novel approach to non-cube optimization with certificates, which handles smooth functions on the hypercube or on the torus. By exploiting the regularity of the target function intrinsic in the decay of its spectrum, we allow at the same time to obtain precise certificates and leverage the advanced and powerful neural networks.
  arXiv  Detail & Related papers  (2023-06-26T09:42:59Z)
• Sparse high-dimensional linear regression with a partitioned empirical Bayes ECM algorithm [62.997667081978825]
  We propose a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates. The proposed approach is implemented in the R package probe.
  arXiv  Detail & Related papers  (2022-09-16T19:15:50Z)
• Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms [9.245687221460654]
  Motivated by applications in single-cell biology and metagenomics, we investigate the problem of matrixing based on a noisy monotone Toeplitz matrix model. We establish fundamental statistical limit for this problem in a decision-theoretic framework and demonstrate that a constrained least squares rate. To address this, we propose a novel-time adaptive sorting algorithm with guaranteed performance improvement.
  arXiv  Detail & Related papers  (2022-01-17T14:53:52Z)
• Bilevel Optimization: Convergence Analysis and Enhanced Design [63.64636047748605]
  Bilevel optimization is a tool for many machine learning problems. We propose a novel stoc-efficientgradient estimator named stoc-BiO.
  arXiv  Detail & Related papers  (2020-10-15T18:09:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.