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The Polar Express: Optimal Matrix Sign Methods and their Application to the Muon Algorithm

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 8.0Download Report PDF
polar decompositionmatrix signnumerical linear algebramuonoptimizationapproximation theory
Abstract:

Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon algorithm for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce Polar Express, a new method for computing the polar decomposition. Like Newton–Schulz and other classical polynomial methods, our approach uses only matrix-matrix multiplications, making it very efficient on GPUs. Inspired by earlier work of Chen & Chow and Nakatsukasa & Freund, Polar Express adapts the update rule at each iteration by solving a minimax optimization problem. We prove that this strategy minimizes error in a worst-case sense, allowing Polar Express to converge as rapidly as possible both in the early iterations and asymptotically. We also address finite-precision issues, making it practical to use in bfloat16. When integrated into Muon, our method yields consistent improvements in validation loss for a GPT-2 model on one to ten billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper introduces Polar Express, an adaptive minimax method for computing polar decomposition tailored to GPU-based neural network training. According to the taxonomy, it resides in the 'Adaptive Minimax Optimization Methods' leaf under 'Polar Decomposition Algorithms and Theory'. Notably, this leaf contains no sibling papers in the current taxonomy, suggesting it occupies a relatively sparse research direction. The taxonomy distinguishes this category from classical fixed-rule methods and continuous-time models, positioning Polar Express as a specialized approach that adapts update rules via minimax optimization for optimal convergence.

The taxonomy reveals that neighboring leaves include 'Classical Numerical Methods' (Newton-based and polynomial schemes), 'Continuous-Time and Discrete-Time Neural Network Models' (zeroing neural networks for time-varying decomposition), and 'Theoretical Foundations and Applications' (mathematical theory and generalizations). The scope notes clarify that Polar Express diverges from classical fixed-rule methods by incorporating adaptive minimax optimization, and from continuous-time models by focusing on discrete iterative updates. The broader 'Neural Network Optimization with Orthogonality Constraints' branch addresses orthogonal training frameworks and low-rank adaptation, but these methods typically enforce constraints rather than compute decompositions as a subroutine.

Among 21 candidates examined across three contributions, none were flagged as clearly refutable. The core algorithm examined 5 candidates with 0 refutable matches; the optimality proof examined 6 candidates with 0 refutable matches; and the finite-precision modifications examined 10 candidates with 0 refutable matches. This suggests that within the limited search scope—top-K semantic matches plus citation expansion—no prior work was found that directly overlaps with the combination of adaptive minimax optimization, GPU-oriented design, and bfloat16 compatibility. The absence of sibling papers in the taxonomy leaf further indicates that this specific intersection of concerns has received limited prior attention.

Based on the limited literature search (21 candidates), the work appears to occupy a novel position at the intersection of classical polar decomposition theory and modern deep learning infrastructure demands. The taxonomy structure and contribution-level statistics suggest that while related methods exist in neighboring leaves, the specific combination of adaptive minimax updates, GPU efficiency, and low-precision arithmetic has not been extensively explored. However, the analysis does not cover exhaustive searches across all numerical linear algebra or optimization venues, leaving open the possibility of relevant work outside the examined scope.

Taxonomy

Core-task Taxonomy Papers
19
3
Claimed Contributions
21
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Computing the polar decomposition for neural network optimization. The field encompasses several distinct branches that reflect both foundational algorithmic concerns and diverse application domains. At its center, Polar Decomposition Algorithms and Theory addresses the numerical methods and convergence guarantees needed to factorize matrices into orthogonal and positive-definite components—a classical problem dating back to foundational work such as Computing Polar Decomposition[5]. Adjacent to this, Neural Network Optimization with Orthogonality Constraints explores how enforcing or learning orthogonal weight matrices can improve training stability and generalization, with recent efforts like Simple Orthogonal Graph[1] and PoLAR[2] demonstrating practical benefits. A third branch, Statistical Estimation and Structured Optimization, examines problems such as group synchronization and optimal transport where polar-like factorizations arise naturally. Meanwhile, Spatial Transformation and Equivariance investigates geometric transformations in vision tasks, and SAR Polarimetry and Remote Sensing Applications focuses on radar signal processing for earth observation, illustrating the breadth of contexts in which polar decompositions prove useful. Within the algorithmic core, a particularly active line of work concerns adaptive minimax optimization methods that balance computational efficiency with numerical stability. Polar Express[0] sits squarely in this space, proposing a novel optimizer that leverages polar decomposition to handle non-convex landscapes more robustly. Its emphasis on adaptive step-size rules and minimax formulations contrasts with earlier approaches like Polar Alignment[3], which focused on aligning learned representations through polar factorization, and PolarGrad[10], which integrates polar updates directly into gradient descent. These neighboring works share a common interest in exploiting orthogonal structure, yet they differ in whether the decomposition is computed explicitly at each iteration or approximated via cheaper surrogates. Open questions remain about the trade-offs between exact polar updates and scalable heuristics, as well as how these methods generalize across different network architectures and loss surfaces.

Claimed Contributions

Polar Express algorithm for computing polar decomposition

The authors propose Polar Express, an iterative method that dynamically adapts polynomial update rules at each iteration by solving a minimax optimization problem. This approach minimizes worst-case error and converges super-exponentially while using only GPU-friendly matrix-matrix multiplications.

5 retrieved papers
Optimality proof for composition of polynomials

The authors prove that their greedy polynomial selection strategy yields the optimal composition of polynomials for approximating the matrix sign function in the supremum norm. This theoretical result (Theorem 3.1) guarantees that Polar Express achieves the best possible worst-case convergence rate.

6 retrieved papers
Finite-precision modifications for bfloat16 compatibility

The authors develop specific modifications to stabilize the algorithm when working in half-precision arithmetic (bfloat16), including rescaling polynomials and using slightly suboptimal polynomials in early iterations to handle numerical round-off errors.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Polar Express algorithm for computing polar decomposition

The authors propose Polar Express, an iterative method that dynamically adapts polynomial update rules at each iteration by solving a minimax optimization problem. This approach minimizes worst-case error and converges super-exponentially while using only GPU-friendly matrix-matrix multiplications.

Contribution

Optimality proof for composition of polynomials

The authors prove that their greedy polynomial selection strategy yields the optimal composition of polynomials for approximating the matrix sign function in the supremum norm. This theoretical result (Theorem 3.1) guarantees that Polar Express achieves the best possible worst-case convergence rate.

Contribution

Finite-precision modifications for bfloat16 compatibility

The authors develop specific modifications to stabilize the algorithm when working in half-precision arithmetic (bfloat16), including rescaling polynomials and using slightly suboptimal polynomials in early iterations to handle numerical round-off errors.

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