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  "title": "Logarithmic alternating-scan mixing for restricted Boltzmann machines with nonnegative interactions and biases",
  "shortTitle": "Two layers, logarithmic mixing",
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  "oneLine": "A uniform logarithmic sweep bound removes a minimum-interaction assumption for nonnegative RBMs; an elementary disjoint-edge example shows the order is necessary.",
  "abstract": "An unrefereed mathematical candidate proves worst-start full-joint total-variation distance at most min{1,(n/2)(2/e)^(2k−2)} after k complete alternating heat-bath sweeps for every finite binary bipartite RBM with finite nonnegative interactions and biases. The constants are independent of graph degree and parameter size. The argument combines an averaged logistic-influence inequality with conditioning before a layer's fresh update. The revised manuscript gives finite-degree sufficient constants, an exact disjoint-edge product reduction, logarithmic-order sharpness and a lattice-qualified cutoff profile for that example. It does not prove optimality of 2/e or general cutoff, and does not cover arbitrary negative biases or signed interactions.",
  "datePublished": "2026-10-14",
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    "name": "Remove the positive minimum-interaction assumption in alternating-scan RBM mixing",
    "url": "https://arxiv.org/abs/2604.00963v2"
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  "keywords": [
    "restricted Boltzmann machines",
    "alternating Gibbs sampler",
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    "A candidate O(log(n/ε)) complete-sweep bound for all finite heterogeneous nonnegative interactions and biases, with absolute constants.",
    "A universal sufficient aggregate logistic-influence coefficient 2/e, applied through fresh-history conditional independence.",
    "An exact binomial full-joint distance reduction and bounded-window cutoff for a disjoint-edge example show logarithmic-order sharpness.",
    "Finite-degree sufficient refinements do not establish optimality of the universal coefficient; a negative-bias obstruction preserves the sign boundary."
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  "relatedWorks": [
    {
      "citation": "Feng, Guo and Yang, Rapid mixing in positively weighted restricted Boltzmann machines, arXiv:2604.00963v2. Theorem 1 and the following minimum-interaction question.",
      "url": "https://arxiv.org/abs/2604.00963v2",
      "doi": "10.48550/arXiv.2604.00963"
    },
    {
      "citation": "Tosh, Mixing Rates for the Alternating Gibbs Sampler over Restricted Boltzmann Machines and Friends, ICML 2016, PMLR 48, 840–849. Alternating-kernel coupling and a norm-restricted RBM mixing theorem.",
      "url": "https://proceedings.mlr.press/v48/tosh16.html",
      "doi": null
    },
    {
      "citation": "Kwon, Qin, Wang and Wei, A phase transition in sampling from restricted Boltzmann machines, Annals of Applied Probability 36(2), 1615–1652 (2026), DOI 10.1214/25-AAP2265. Appendix A Lemma 21 refers specifically to the linked arXiv version 1.",
      "url": "https://arxiv.org/abs/2410.08423v1",
      "doi": "10.48550/arXiv.2410.08423"
    },
    {
      "citation": "Chen and Kumagai, Cutoffs for product chains, Stochastic Processes and their Applications128(11), 3840–3879 (2018). Product-chain context; update-clock constants are not imported.",
      "url": "https://doi.org/10.1016/j.spa.2018.01.002",
      "doi": "10.1016/j.spa.2018.01.002"
    },
    {
      "citation": "Chin, Moitra, Mossel and Sandon, The power of an adversary in Glauber dynamics, COLT2024, PMLR247, 1102–1124. Methodological context for strong-field concavity, not the uniform alternating-RBM theorem.",
      "url": "https://proceedings.mlr.press/v247/chin24a.html",
      "doi": null
    }
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