{
  "abstract": "Dictionary learning seeks to recover an unknown dictionary $A$ from observations ${\\bf y}_i = A{\\bf x}_i$ with sparse coefficient vectors ${\\bf x}_i$. We introduce the \\emph{Iterative Atom Refinement} (IAR) algorithm, a simple procedure for recovering individual dictionary atoms. Starting from a random direction, IAR repeatedly selects the observations most strongly correlated with the current iterate and updates the direction by averaging the selected data. Our main contribution is a rigorous convergence theory of IAR. Using high-dimensional probabilistic estimates and a novel monotonicity principle for atom-selection probabilities, we show that a small initial advantage of one atom is amplified until that atom is isolated. Under our model assumptions, IAR identifies a generating atom after only three refinement steps. Numerical experiments support the theory and show that the resulting dynamics accurately capture the behavior observed in dictionary refinement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23812",
  "author_total": 5,
  "authors": [
    {
      "name": "Alexander Christie"
    },
    {
      "name": "Miguel Moscoso"
    },
    {
      "name": "Alexei Novikov"
    },
    {
      "name": "George Papanicolaou"
    },
    {
      "name": "Chrysoula Tsogka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11yssa3y5kvbm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23812",
  "fulltext_url": null,
  "id": "11yssa3y5kvbm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23812",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7214038513",
  "pdf_url": "https://arxiv.org/pdf/2609.23812",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11yssa3y5kvbm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23812",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23812"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Iterative Atom Refinement: A Monotonicity Principle for Dictionary Learning",
  "title_text": "Iterative Atom Refinement: A Monotonicity Principle for Dictionary Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:33Z",
  "url": "https://research.draconitic.com/paper/11yssa3y5kvbm",
  "venue": "arXiv",
  "versions": []
}