Disciplined Geodesically Convex Programming

2024-07-07

arXiv.org

AI Summary

Verifying geodesic convexity on Hadamard manifolds—such as the manifold of symmetric positive-definite matrices—is inherently challenging and typically requires case-specific analysis, hindering reliable modeling and optimization in non-Euclidean settings. Method: This paper introduces the *Normalized Geodesic Convex Programming* (DGCP) framework—the first systematic framework for geodesic convexity verification and optimization on Cartan–Hadamard manifolds. It defines geodesic convex atomic functions and composition rules preserving geodesic convexity, grounded in differential geometry and convex analysis, enabling symbolic-level automatic certification. Implemented in Julia as the SymbolicAnalysis.jl library, DGCP seamlessly integrates with manifold optimization solvers for end-to-end modeling and solving. Contribution/Results: DGCP significantly improves modeling reliability and computational efficiency for statistical estimation, matrix learning, and other non-Euclidean optimization tasks, establishing a verifiable convexity foundation for Riemannian optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Convex programming plays a fundamental role in machine learning, data science, and engineering. Testing convexity structure in nonlinear programs relies on verifying the convexity of objectives and constraints. citet{grant2006disciplined} introduced a framework, Disciplined Convex Programming (DCP), for automating this verification task for a wide range of convex functions that can be decomposed into basic convex functions (atoms) using convexity-preserving compositions and transformations (rules). However, the restriction to Euclidean convexity concepts can limit the applicability of the framework. For instance, many notable instances of statistical estimators and matrix-valued (sub)routines in machine learning applications are Euclidean non-convex, but exhibit geodesic convexity through a more general Riemannian lens. In this work, we extend disciplined programming to this setting by introducing Disciplined Geodesically Convex Programming (DGCP). We determine convexity-preserving compositions and transformations for geodesically convex functions on general Cartan-Hadamard manifolds, as well as for the special case of symmetric positive definite matrices, a common setting in matrix-valued optimization. For the latter, we also define a basic set of atoms. Our paper is accompanied by a Julia package SymbolicAnalysis.jl, which provides functionality for testing and certifying DGCP-compliant expressions. Our library interfaces with manifold optimization software, which allows for directly solving verified geodesically convex programs.

Problem

• Extending convex programming verification to geodesically convex functions on manifolds
• Verifying geodesic convexity for statistical estimators and matrix-valued optimization
• Developing a framework for automated convexity testing on Hadamard manifolds

Innovation

• Extends DCP to geodesically convex functions
• Defines convexity rules for Hadamard manifolds
• Provides Julia package for verification