Disciplined Geodesically Convex Programming

TL;DR

This paper extends the disciplined convex programming framework to geodesically convex functions on manifolds, enabling automated verification and solution of a broader class of convex optimization problems in Riemannian geometry.

Contribution

We introduce Disciplined Geodesically Convex Programming (DGCP), expanding convexity verification to functions on Hadamard manifolds and matrix manifolds, with a supporting Julia package.

Findings

DGCP framework for geodesic convexity on manifolds.

A Julia package for testing and certifying DGCP expressions.

Integration with manifold optimization software for solving problems.

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. Grant et al. (2006) 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). Here, we extend this framework to functions defined on manifolds with non-positive curvature (Hadamard manifolds) by introducing Disciplined Geodesically Convex Programming (DGCP). In particular, this allows for verifying a broader range of convexity notions. 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. To define the DGCP framework, we determine convexity-preserving compositions and transformations for geodesically convex functions on general 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.