{
  "abstract": "Can independent components be recovered when all of them are Gaussian? Yes, if their variances change across environments; but how many environments, and how many samples as the number of components $d$ grows? We answer both for the joint diagonalization of $K$ covariances. Each component has a signature, its variances across the environments. Separating component $j$ costs the Riesz 2-energy $E_j$ of this signature, a sum, over the other components, of inverse pairwise Fisher informations. For Gaussian sources no estimator in $O(d)$ beats $cE_j/(n+E_j)$ at any $n$ and $d$, and above an explicit sample size a spectral estimator attains $E_j/n$ up to a constant, for every mixing and without the signatures (given a lower bound on the pairwise information). Since signatures live in $K-1$ dimensions, the rate obeys a dimension law: $d^2/n$, $d\\log d/n$ and $d/n$ for $K=2$, $3$ and $K\\ge4$ well-spread environments. What counts is the dimension the signatures span, not their number, and $\\log d$ well-designed environments make $d\\log d$ samples suffice. Projecting the covariances onto one linear combination pays $d^2/n$ at first order instead. In simulations up to $d=1000$ the predictions hold and joint diagonalization with $K\\ge4$ attains the rate once $n/d$ is in the hundreds; on task fMRI the energy says which components are hard to separate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06228",
  "author_total": 6,
  "authors": [
    {
      "name": "Yuda Bi"
    },
    {
      "name": "Yubo Guo"
    },
    {
      "name": "Jiaxiao Cai"
    },
    {
      "name": "Xiaonai Li"
    },
    {
      "name": "Zhuolin Li"
    },
    {
      "name": "Vince D. Calhoun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1n5gb63bh4rcy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06228",
  "fulltext_url": null,
  "id": "1n5gb63bh4rcy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06228",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06228",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1n5gb63bh4rcy.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06228",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06228"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Separation Is a Riesz Energy: How Many Environments Does ICA Need?",
  "title_text": "Separation Is a Riesz Energy: How Many Environments Does ICA Need?",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:32Z",
  "url": "https://research.draconitic.com/paper/1n5gb63bh4rcy",
  "venue": "arXiv",
  "versions": []
}