Beyond Mutual Information: Extension Profiles and Shape Functions of Random Variable Pairs
Abstract
We study the extension profile of a pair of jointly distributed finite-valued random variables (X,Y), defined as the set of all triples of numbers (H(X|W),H(Y|W),I(X:Y|W)) obtained by extending the pair with an auxiliary random variable W. This object captures structural properties of a joint distribution that are not determined by the entropies of X and Y and their mutual information We associate with the extension profile its shape function defined in terms of the Legendre--Fenchel transform of the profile's lower boundary. We establish upper and lower bounds on the shape function in terms of classical information-theoretic quantities. For random pairs (X,Y) whose joint distribution is uniform on its support, we relate the shape function to combinatorial and spectral properties of the associated bipartite support graph. We show that for random pairs uniformly distributed on the edges of balanced expander graphs, the shape function is within O(logH(XY)) of the pointwise infimum over all pairs, which proves the conjecture stated in [MR26]. These results connect the extension profile with non-Shannon-type information inequalities, Gács--Körner common information, and spectral graph theory. We apply these techniques to study an approximate version of a question posed by Zhen Zhang ([Zha03]) concerning the characterization of pairs with maximally non-extractable mutual information.
Related
- Maximum-Entropy Extension of Quantum Correlation Functions from Short Real-Time Dynamics
- Continuous variable quantum key distribution through a fading channel with minimal information leakage
- The Regularized Frozen Natural Orbital approach to CASPT2: extension to Restricted Active Space wave functions, different multiplicities and assessment of correlation effects
- Extension of a multi-region free-surface MHD solver beyond the inductionless approximation
- Can future reirradiation justify upfront proton therapy? An extension of model-based proton therapy selection beyond primary-course NTCP benefit
BibTeX
@misc{matveev2026beyond,
title = {{Beyond Mutual Information: Extension Profiles and Shape Functions of Random Variable Pairs}},
author = {Rostislav Matveev and Andrei Romashchenko},
howpublished = {HAL},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.23849},
eprint = {2606.23849},
archivePrefix = {arXiv},
url = {https://hal-lirmm.ccsd.cnrs.fr/lirmm-05667618v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond Mutual Information: Extension Profiles and Shape Functions of Random Variable Pairs
AU - Rostislav Matveev
AU - Andrei Romashchenko
PY - 2026
DA - 2026/06/22
PB - HAL
DO - 10.48550/arxiv.2606.23849
UR - https://hal-lirmm.ccsd.cnrs.fr/lirmm-05667618v1
AB - We study the extension profile of a pair of jointly distributed finite-valued random variables (X,Y), defined as the set of all triples of numbers (H(X|W),H(Y|W),I(X:Y|W)) obtained by extending the pair with an auxiliary random variable W. This object captures structural properties of a joint distribution that are not determined by the entropies of X and Y and their mutual information We associate with the extension profile its shape function defined in terms of the Legendre--Fenchel transform of the profile's lower boundary. We establish upper and lower bounds on the shape function in terms of classical information-theoretic quantities. For random pairs (X,Y) whose joint distribution is uniform on its support, we relate the shape function to combinatorial and spectral properties of the associated bipartite support graph. We show that for random pairs uniformly distributed on the edges of balanced expander graphs, the shape function is within O(logH(XY)) of the pointwise infimum over all pairs, which proves the conjecture stated in [MR26]. These results connect the extension profile with non-Shannon-type information inequalities, Gács--Körner common information, and spectral graph theory. We apply these techniques to study an approximate version of a question posed by Zhen Zhang ([Zha03]) concerning the characterization of pairs with maximally non-extractable mutual information.
ER -