{
  "abstract": "We develop a novel approach to matrix discrepancy based on matrix small-ball estimates. Specifically, we use a determinantal weight (obtained from the log-barrier) to scale the small-ball probability into a partition function of a tilt of the Gaussian measure. We then employ matrix-weighted Poincaré inequalities to compare this partition function to that of a pinched or diagonal part of the matrix, obtaining \\emph{dimension-free} constants. Our technique yields a hereditary small-ball estimate for Gaussian series that should be of independent interest. As the main application, we resolve the Matrix Spencer conjecture: for symmetric $n\\times n$ matrices $A_1,\\dots,A_n$ with $\\|A_i\\|\\le1$, one can efficiently find a coloring $x\\in\\{\\pm1\\}^n$ with $\\|\\sum_{i=1}^n x_iA_i\\|=O(\\sqrt n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28816",
  "author_total": 2,
  "authors": [
    {
      "name": "Emrullah Akbas"
    },
    {
      "name": "Suvrit Sra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a3perjkrr3gqy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28816",
  "fulltext_url": null,
  "id": "a3perjkrr3gqy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28816",
  "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",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7204899715",
  "pdf_url": "https://arxiv.org/pdf/2608.28816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a3perjkrr3gqy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Proof of the Matrix Spencer Conjecture",
  "title_text": "A Proof of the Matrix Spencer Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/a3perjkrr3gqy",
  "venue": "arXiv",
  "versions": []
}