Pith Number

pith:B6V3D5GU

pith:2026:B6V3D5GUS4CF3TKN3JEMLBC7CY

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A Global Characterization of $f$-Divergences Yielding PSD Mutual-Information Matrices

Zachary Robertson

Mutual-information matrices from f-divergences are positive semidefinite for all finite alphabets precisely when the normalized generator expands as a power series with nonnegative coefficients that converges on all positive reals.

arxiv:2601.08929 v3 · 2026-01-13 · cs.IT · math.IT

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4 Citations open

5 Replications open

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Claims

C1strongest claim

The matrix M^{(f)}_{ij}:=I_f(X_i;X_j) is PSD for every finite-alphabet family if and only if the normalized representative has a globally convergent expansion bar f(t)=sum_{m>=2} a_m (t-1)^m, with a_m >=0, on all of (0,infty).

C2weakest assumption

That the local positivity condition at t=1, extracted via biased three-point kernels and the BGKP theorem, extends to global analyticity and holds for all finite alphabets without additional restrictions on the divergence.

C3one line summary

Pairwise f-mutual information matrices are positive semi-definite for all finite-alphabet distributions exactly when the f generator has a power series with all nonnegative coefficients that converges on the positive reals.

References

15 extracted · 15 resolved · 0 Pith anchors

[1] The mutual information: detecting and evaluating dependencies between variables, 2002

[2] Detecting novel associations in large data sets, 2011

[3] G. Ver Steeg and A. Galstyan, “The information sieve,” inInternational Conference on Machine Learning. PMLR, 2015, pp. 164–172 2015

[4] How transformers learn causal structure with gradient descent 2024

[5] Mutual information matrices are not always positive semidefinite, 2014

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:00.248211Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

0fabb1f4d497045dcd4dda48c5845f163e99b9d05254cc2b05e35a70e436fc8f

Aliases

arxiv: 2601.08929 · arxiv_version: 2601.08929v3 · doi: 10.48550/arxiv.2601.08929 · pith_short_12: B6V3D5GUS4CF · pith_short_16: B6V3D5GUS4CF3TKN · pith_short_8: B6V3D5GU

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/B6V3D5GUS4CF3TKN3JEMLBC7CY \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 0fabb1f4d497045dcd4dda48c5845f163e99b9d05254cc2b05e35a70e436fc8f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "76cb0e1841cf9733024256918b40c680caa7ba07b955a674a95f70c6b049b040",
    "cross_cats_sorted": [
      "math.IT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.IT",
    "submitted_at": "2026-01-13T19:09:19Z",
    "title_canon_sha256": "9f1484ceff67644f2fe001cef2f926c4a94c0a13fcd71ee32ed1e023b45a1795"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.08929",
    "kind": "arxiv",
    "version": 3
  }
}