On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines
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 spectra...
| Authors | Chao Li, Zerui Tao, Yuchen Cong et al. |
|---|---|
| Primary category | cs.LG |
| Published | 12 May 2026 |
| Last updated at source | 9 October 2026 |
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