Polylab: A MATLAB Toolbox for Multivariate Polynomial Modeling

2026-04-07

AI Summary

This work addresses the lack of unified CPU/GPU backend support and symbolic-numeric integration in existing tools for multivariate polynomial modeling. We propose a MATLAB-based toolbox offering three interfaces—MPOLY, MPOLY_GPU, and MPOLY_HP—that enable construction, algebraic manipulation, differentiation, LaTeX export, and interoperability with YALMIP and SOSTOOLS for polynomials and polynomial matrices. Key innovations include a variable metadata mechanism ensuring safe handling of mixed-variable expressions, a dedicated layer for affine normal vector computation, and a unified CPU/GPU architecture that seamlessly blends symbolic and numeric processing. Experimental results demonstrate that MPOLY is well-suited for lightweight interactive tasks, MPOLY_HP achieves higher efficiency in medium- to large-scale affine computations, and the proposed randomized log-determinant algorithm exhibits significant advantages in sparse, large-scale settings.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Polylab is a MATLAB toolbox for multivariate polynomial scalars and polynomial matrices with a unified symbolic-numeric interface across CPU and GPU-oriented backends. The software exposes three aligned classes: MPOLY for CPU execution, MPOLY_GPU as a legacy GPU baseline, and MPOLY_HP as an improved GPU-oriented implementation. Across these backends, Polylab supports polynomial construction, algebraic manipulation, simplification, matrix operations, differentiation, Jacobian and Hessian construction, LaTeX export, CPU-side LaTeX reconstruction, backend conversion, and interoperability with YALMIP and SOSTOOLS. Versions 3.0 and 3.1 add two practically important extensions: explicit variable metadata through vars.id and vars.name, which makes mixed-variable expressions safe even when objects are created independently, and affine-normal direction computation via automatic differentiation, MF-logDet-Exact, and MF-logDet-Stochastic. The toolbox has already been used successfully in prior research applications, and Polylab Version 3.1 adds a new geometry-oriented computational layer on top of a mature polynomial modeling core. This paper documents the architecture and user-facing interface of the software, organizes its functionality by workflow, presents representative MATLAB sessions with actual outputs, and reports reproducible benchmarks. The results show that MPOLY is the right default for lightweight interactive workloads, whereas MPOLY-HP becomes advantageous for reduction-heavy simplification and medium-to-large affine-normal computation; the stochastic log-determinant variant becomes attractive in larger sparse regimes under approximation-oriented parameter choices.

Problem

• multivariate polynomial modeling
• symbolic-numeric computation
• CPU-GPU interoperability
• affine-normal computation
• polynomial matrix operations

Innovation

• multivariate polynomial modeling
• symbolic-numeric interface
• GPU acceleration
• variable metadata
• affine-normal computation