{
  "abstract": "We study when statistically learnable latent structure can also be recovered efficiently, and how membership queries change the answer. An unknown support $A\\subseteq\\mathbb F_2^n$ has small additive doubling and is observed through a fixed set $B$ satisfying $|A\\triangle B|\\leη|A|$. We seek one linear subspace $V$ such that every compatible support $A$ is covered by few $V$-cosets and satisfies $|V|\\le|A|$. For every $η<1$, polynomially many uniform samples suffice statistically, with cost polynomial in the doubling constant and proportional to $(1-η)^{-1}$; this radius dependence is sharp. Under a specified hardness assumption for learning parities with noise (search-LPN), however, no polynomial-time sample-only learner achieves even constant covering cost, including when the latent support is unique. At fixed structural parameters and the same constant covering budget, adding exact membership queries to $B$ permits polynomial-time recovery. The general query learner constructs a short structural list and uses fresh samples to select one common output through a majority-coverage rule. Persistent structured cores make this candidate construction possible. At doubling one, a complementary distinction appears at $η=1/3$: coarse recovery remains polynomial time, while exact recovery requires exponentially many accesses in the worst case when latent cardinality is unknown.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00451",
  "author_total": 1,
  "authors": [
    {
      "name": "Cheng Peng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdmzv3r6nvw6j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00451",
  "fulltext_url": null,
  "id": "fdmzv3r6nvw6j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00451",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7172461572",
  "pdf_url": "https://arxiv.org/pdf/2608.00451",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdmzv3r6nvw6j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2608.00451",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00451"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Learning Latent Algebraic Structure from Ambiguous Set Observations",
  "title_text": "Learning Latent Algebraic Structure from Ambiguous Set Observations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/fdmzv3r6nvw6j",
  "venue": "arXiv",
  "versions": []
}