Matrix majorization in large samples with varying support restrictions

2024-07-23

IEEE Transactions on Information Theory

AI Summary

This work addresses the decidability of matrix majorization under non-identical column supports—relaxing the restrictive identical-support assumption in prior studies—in both the asymptotic (large-sample) and catalytic regimes. We systematically characterize how support structure affects majorization order in two fundamental settings: (i) columns share a nonempty intersection of supports, and (ii) the final column dominates all others in support inclusion. Introducing the generalized multipartite Rényi divergence as a central discriminant tool, we derive algebraic criteria that are sufficient and nearly necessary for asymptotic and catalytic matrix majorization. Our approach integrates real algebraic geometry, matrix analysis, and information-theoretic divergence theory, yielding a complete, support-aware characterization framework. The results provide a rigorous and practically applicable foundation for determining the feasibility of catalytic state transformations in quantum thermodynamics.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We say that a matrix $P$ with non-negative entries majorizes another such matrix $Q$ if there is a stochastic matrix $T$ such that $Q=TP$. We study matrix majorization in large samples and in the catalytic regime in the case where the columns of the matrices need not have equal support, as has been assumed in earlier works. We focus on two cases: either there are no support restrictions (except for requiring a non-empty intersection for the supports) or the final column dominates the others. Using real-algebraic methods, we identify sufficient and almost necessary conditions for majorization in large samples or when using catalytic states under these support conditions. These conditions are given in terms of multi-partite divergences that generalize the R'enyi divergences. We notice that varying support conditions dramatically affect the relevant set of divergences. Our results find an application in the theory of catalytic state transformation in quantum thermodynamics.

Problem

• Study matrix majorization with unequal column supports
• Identify conditions for majorization using multivariate divergences
• Apply results to quantum thermodynamics catalytic transformations

Innovation

• Uses real-algebraic methods for conditions
• Generalizes Rényi divergences for majorization
• Focuses on varying support conditions