{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0013",
  "title": "Critical Points and Local Maxima of Sparse Binary Restricted Boltzmann Machines",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We classify the finite critical points of exact empirical log likelihood for binary restricted Boltzmann machines in which every hidden unit has at most two visible neighbors. All allowed weights and biases are independent real parameters. A pairwise exponential-family representation and its singularity analysis give the exact Hessian inertia, no spurious finite local maxima, and strict saddles at all finite nonglobal critical points. We describe finite attainment, critical visible distributions through active-edge submodels, and regular and singular maximizing fibers. Two six-observation samples show that sample size and pairwise support patterns do not decide finite attainment. A three-input hidden unit supplies a nonstrict saddle, while a four-input hidden unit supplies a finite spurious local maximum for full-support parity-biased data, with an explicit finite point of strictly higher likelihood. These scoped theorems concern AIM-PROBABILITY-0013 (AIM Boltzmann Machines Problem 5.2); they do not classify arbitrary RBM architectures or close the general source question. Standard softplus and exponential-family results, prior RBM likelihood work, and the inherited leaf-hidden special case are credited. This AI-assisted preprint is self-audited and unrefereed; novelty and absolute priority are not certified.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.ST",
    "stat.ML"
  ],
  "keywords": [
    "restricted Boltzmann machines",
    "likelihood landscape",
    "Hessian inertia",
    "exponential families",
    "spurious local maxima",
    "AIM-PROBABILITY-0013"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0013/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0013/paper.pdf?v=3efd9021cb6c",
  "doi": "10.5281/zenodo.23171795",
  "zenodo_record_url": "https://zenodo.org/records/23171795",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete finite critical-point classification for independent-real-parameter binary RBMs with hidden degrees at most two, together with explicit degree-three and degree-four limitations. The arbitrary-architecture characterization requested by AIM Boltzmann Machines Problem 5.2 remains open. The written general proofs are supplemented by 65 exact checks, not replaced by them. Standard representation and exponential-family theory, prior RBM likelihood work, and the inherited leaf-hidden special case are credited. AI-assisted, self-audited and unrefereed; no independent review or proof-assistant formalization is claimed. Novelty and absolute priority remain undetermined.",
  "files": {
    "paper.pdf": {
      "sha256": "3efd9021cb6c9469c46507196a0ae856c7317601408f98d19fd42db06cb68a35"
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    "source.zip": {
      "sha256": "c9af7839a4fdf13e1b058410eb7fb8f25904034d294dd1cf2c18841bb587d1fc"
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    "verification_report.md": {
      "sha256": "3559264fb1514b248bc0bd312e98f4d08955b73c0ec84c660d3fc4bd94882146"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}