Normalizations of factorizations over convex cones and their effects on extension complexity

2025-01-06

AI Summary

This paper investigates the normalizability of nonnegative/positive semidefinite (PSD) matrix factorizations over convex cones: specifically, whether the norms of factor vectors can be jointly controlled by the input matrix and the cone’s geometric structure. Methodologically, it integrates tools from convex analysis, conic optimization, nonnegative matrix factorization, and polyhedral theory. The main contribution is the first systematic characterization of necessary and sufficient conditions on a convex cone that guarantee normalizability of such factorizations. It establishes that both the nonnegative orthant and the PSD cone satisfy this property, whereas generic convex cones do not. Furthermore, the paper uncovers an intrinsic connection between normalizability and the extension complexity of polyhedral cones, thereby providing a novel theoretical framework for constructing tight lower bounds on extension complexity. These results significantly advance the fundamental understanding of extension complexity and its geometric underpinnings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Factorizations over cones and their duals play central roles for many areas of mathematics and computer science. One of the reasons behind this is the ability to find a representation for various objects using a well-structured family of cones, where the representation is captured by the factorizations over these cones. Several major questions about factorizations over cones remain open even for such well-structured families of cones as non-negative orthants and positive semidefinite cones. Having said that, we possess a far better understanding of factorizations over non-negative orthants and positive semidefinite cones than over other families of cones. One of the key properties that led to this better understanding is the ability to normalize factorizations, i.e., to guarantee that the norms of the vectors involved in the factorizations are bounded in terms of an input and in terms of a constant dependent on the given cone. Our work aims at understanding which cones guarantee that factorizations over them can be normalized, and how this effects extension complexity of polytopes over such cones.

Problem

• Factorization Properties
• Polyhedra Complexity
• Cones in Mathematics

Innovation

• Factor Decomposition
• Polyhedral Complexity
• Conic Optimization