On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines

Abstract:Matrix product operator Born machines (MPO-BMs) are tractable tensor-network models for probabilistic modeling, yet their approximation capabilities remain poorly understood in theory. In this paper, we characterize these capabilities from both negative and positive perspectives. As a negative result, we prove that constructing a polynomial-bond MPO-BM approximation to an arbitrary continuous target with bounded KL error is NP-hard. On the positive side, for score-based variational inference, we prove that targets satisfying a log-Sobolev inequality admit polynomial-bond MPO-BM approximations with provable KL guarantees when the loss-induced Hamiltonian is local and has a constant spectral gap. Furthermore, under the same conditions, we construct local importance-sampling estimators and prove that polynomially many local score queries suffice to preserve the KL guarantees.

Machine Learning (cs.LG)

arXiv:2605.11471 [cs.LG]

(or arXiv:2605.11471v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.11471 Focus to learn more arXiv-issued DOI via DataCite

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