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

arXiv 2605.11471 ↗First posted 12 May 2026cs.LGcited by 0

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Abstract (arXiv)

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.

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References · 25

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13 more
  1. 2024What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?cited by 22
  2. 2021Tensor networks for unsupervised machine learningcited by 21
  3. 2024Sum of Squares Circuitscited by 21
  4. 2023Variational Inference with Gaussian Score Matchingcited by 17
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  6. 2024EigenVI: score-based variational inference with orthogonal function expansionscited by 11
  7. 2023Generative Learning of Continuous Data by Tensor Networkscited by 9
  8. 2023Generative Modeling via Hierarchical Tensor Sketchingcited by 8
  9. 2023TERM Model: Tensor Ring Mixture Model for Density Estimationcited by 2
  10. 2024Using matrix-product states for time-series machine learningcited by 2
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  12. 2025On the Hardness of Approximating Distributions with Tractable Probabilistic Modelscited by 2
  13. 2025Regularized second-order optimization of tensor-network Born machinescited by 1

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