Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Cheng, Andrew, Dixit, Vaibhav, Weber, Melanie
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funders: National Science Foundation, U.S. Department of Energy, Equinor, Norges Forskningsråd, Advanced Research Projects Agency - Energy, Advanced Research Projects Agency, Natural Sciences and Engineering Research Council of Canada, Division of Chemical, Bioengineering, Environmental, and Transport Systems, Division of Electrical, Communications and Cyber Systems
Awards: G1467515376, G1498456215, eMB: Collaborative Research: Discovery and calibration of stochastic chemical reaction network models, SWIFT: LARGE: Adaptive Radio Frequency Interference Cancellation for Radio Science Observatories, G4536555042, G6553639979, Collaborative Research: Frameworks: Convergence of Bayesian inverse methods and scientific machine learning in Earth system models through universal differentiable programming, AI Institute in Dynamic Systems, G7146022962, Framework: Software: Next-Generation Cyberinfrastructure for Large-Scale Computer-Based Scientific Analysis and Discovery, G7678611020, G7789225813, THIS COOPERATIVE AGREEMENT IS AWARDED TO CARNEGIE MELLON UNIVERSITY FOR A PROJECT ENTITLED “HIGH-FIDELITY ACCELERATED DESIGN OF HIGH-PERFORMANCE ELECTROCHEMICAL SYSTEMS”. THIS AWARD IS BEING MADE IN ACCORDANCE WITH FOA NUMBER DE-FOA-0002107 DIFFERENTIATE AND APPLICATION CONTROL NUMBER 2107-1548., ACCELERATING COUPLED HVAC BUILDING SIMULATION WITH A NEURAL COMPONENT ARCHITECTURE
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