Variational Inference on the Boolean Hypercube with the Quantum Entropy

In this paper, we derive variational inference upper-bounds on the log-partition function of pairwise Markov random fields on the Boolean hypercube, based on quantum relaxations of the Kullback-Leibler divergence. We then propose an efficient algorithm to compute these bounds based on primal-dual optimization. An improvement of these bounds through the use of ''hierarchies,'' similar to sum-of-squares (SoS) hierarchies is proposed, and we present a greedy algorithm to select among these relaxations. We carry extensive numerical experiments and compare with state-of-the-art methods for this inference problem.

@article{beyler2025_2411.03759,
  title={ Variational Inference on the Boolean Hypercube with the Quantum Entropy },
  author={ Eliot Beyler and Francis Bach },
  journal={arXiv preprint arXiv:2411.03759},
  year={ 2025 }
}