Near-Linear Runtime for a Classical Matrix Preconditioning Algorithm

2025-03-20

AI Summary

This work resolves a long-standing theoretical gap in Osborne’s matrix balancing algorithm: despite its empirical efficiency and widespread adoption in mainstream numerical software, its original formulation only admits polynomial-time complexity, while existing near-linear-time guarantees rely on variants that are empirically slower. We establish the first near-linear time complexity upper bound—$ ilde{O}(mathrm{nnz}(A))$—for the original Osborne algorithm, matching the input size up to logarithmic factors and outperforming downstream tasks such as eigenvalue computation. Technically, our analysis integrates potential function methods, fixed-point iteration theory, and precise error propagation control. This result bridges a more than decade-old disconnect between theoretical analysis and industrial practice in matrix balancing.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Algorithm Configuration

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In 1960, Osborne proposed a simple iterative algorithm for matrix balancing with outstanding numerical performance. Today, it is the default preconditioning procedure before eigenvalue computation and other linear algebra subroutines in mainstream software packages such as Python, Julia, MATLAB, EISPACK, LAPACK, and more. Despite its widespread usage, Osborne's algorithm has long resisted theoretical guarantees for its runtime: the first polynomial-time guarantees were obtained only in the past decade, and recent near-linear runtimes remain confined to variants of Osborne's algorithm with important differences that make them simpler to analyze but empirically slower. In this paper, we address this longstanding gap between theory and practice by proving that Osborne's original algorithm -- the de facto preconditioner in practice -- in fact has a near-linear runtime. This runtime guarantee (1) is optimal in the input size up to at most a single logarithm, (2) is the first runtime for Osborne's algorithm that does not dominate the runtime of downstream tasks like eigenvalue computation, and (3) improves upon the theoretical runtimes for all other variants of Osborne's algorithm.

Problem

• Proves near-linear runtime for Osborne's matrix balancing algorithm.
• Bridges theory-practice gap in matrix preconditioning runtime guarantees.
• Optimizes runtime to not dominate downstream eigenvalue computation tasks.

Innovation

• Proves Osborne's algorithm has near-linear runtime
• Optimal runtime up to single logarithm
• Improves theoretical runtimes for all variants