{
  "abstract": "Graph-data workloads such as diffusion estimation, ranking, semi-supervised learning, and network optimization often solve many Laplacian or symmetric diagonally dominant M-matrix (SDDM) systems with the same coefficient matrix. Approximate Cholesky preconditioners eliminate vertices one at a time and store the resulting sparse approximate factorization, the \\emph{factor}, whose construction cost is amortized across these solves. But eliminating a vertex, the \\emph{pivot}, creates a dense Schur-complement clique among its $d$ active neighbors. We introduce CAST (Canonical Approximate Schur Tree), which replaces this clique with a weighted random spanning tree sampled directly from it. Every realization is connected and contains exactly d-1 edges, while reweighting each selected edge by the reciprocal of its tree-inclusion probability makes the update unbiased. The distribution is independent of the ordering of the pivot neighbors, and we prove that its leverage-score marginals minimize the largest normalized reweighted-edge contribution among unbiased inverse-marginal one-tree estimators. We also introduce CAST-$ρ$, which replaces each pivot neighbor with $ρ$ copies, each carrying a $1/ρ$ share of that neighbor's incident weight, samples a weighted random spanning tree on the expanded clique, and contracts the copies back to the original neighborhood. The resulting update remains unbiased and connected, can be sampled exactly in $O(ρd)$ time, and satisfies a $1/ρ$ bound on the second moment of the normalized local Schur error. Increasing $ρ$ therefore reduces certified local sampling variability, but may increase construction cost and downstream fill. Empirically, we observe that CAST-1 is the faster default, whereas CAST-2 is preferable when its additional edge contributions remain inexpensive.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09255",
  "author_total": 3,
  "authors": [
    {
      "name": "Meher Chaitanya"
    },
    {
      "name": "Cameron Musco"
    },
    {
      "name": "Aristides Gionis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bc3ph60datjtn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09255",
  "fulltext_url": null,
  "id": "bc3ph60datjtn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09255",
  "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.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212125880",
  "pdf_url": "https://arxiv.org/pdf/2609.09255",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bc3ph60datjtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09255",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09255"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "CAST: Canonical Approximate Schur Tree for Approximate Cholesky on Graphs",
  "title_text": "CAST: Canonical Approximate Schur Tree for Approximate Cholesky on Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:22Z",
  "url": "https://research.draconitic.com/paper/bc3ph60datjtn",
  "venue": "arXiv",
  "versions": []
}