Pith Number

pith:TAHL4WII

pith:2026:TAHL4WIIHB73FBYDMFNJJ6AW5R

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Bounds on the Number of Modes of a Gaussian Mixture Density

Hien Duy Nguyen

Gaussian mixture densities with k components have at most floor of (min of two algebraic bounds plus one) divided by two modes when the modal set is finite.

arxiv:2605.15531 v1 · 2026-05-15 · math.ST · math.CO · stat.TH

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Claims

C1strongest claim

For k≥2, the direct Pfaffian bound is U_het(d,k)=2^{d+binom(k-1,2)}(d+2 min(d,k-1)+1)^{k-1}, with the best critical-point bound being the minimum of this and the augmented bound, and the finite-mode bound improved by Morse theory to floor((min{U_het,U_aug}+1)/2).

C2weakest assumption

The critical-point equations can be normalized by a reference component without loss of generality for k≥2, and the Morse-theoretic argument applies directly to improve the finite-mode upper bound when the modal set is finite.

C3one line summary

Explicit upper bounds on nondegenerate critical points of k-component Gaussian mixture densities are given via Pfaffian and algebraic elimination methods, with homoscedastic simplifications and combinatorial lower bounds.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] Alexandrovich, G., Holzmann, H., and Ray, S. (2013). On the number of modes of finite mixtures of elliptical distributions. In B. Lausen, D. van den Poel, and A. Ultsch (Eds.),Algorithms from and for 2013 · doi:10.1093/imaiai/iaz013

[2] Cheng, Y.(1995).Meanshift, modeseeking, andclustering.IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(8), 790–799. 34 1995

[3] Ester, M., Kriegel, H.-P., Sander, J., and Xu, X. (1996). A density-based algorithm for discov- ering clusters in large spatial databases with noise. InProceedings of the Second International Conferen 1996

[4] Gabrielov, A., and Vorobjov, N. (2004). Complexity of computations with Pfaffian and Noetherian functions. InNormal Forms, Bifurcations and Finiteness Problems in Differential Equations (pp. 211–250). 2004

[5] Kabata, Y., Matsumoto, H., Uchida, S., and Ueki, M. (2025). Singularities in bivariate normal mixtures.Information Geometry, 8, 343–357 2025

Receipt and verification

First computed	2026-05-20T00:01:03.687197Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

980ebe5908387fb28703615a94f816ec4809a42ac93454b5d985e2073d297d00

Aliases

arxiv: 2605.15531 · arxiv_version: 2605.15531v1 · doi: 10.48550/arxiv.2605.15531 · pith_short_12: TAHL4WIIHB73 · pith_short_16: TAHL4WIIHB73FBYD · pith_short_8: TAHL4WII

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/TAHL4WIIHB73FBYDMFNJJ6AW5R \
  | 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: 980ebe5908387fb28703615a94f816ec4809a42ac93454b5d985e2073d297d00

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "04cda0b859209116bf667dc4c05f72f30dc4eecc8e5058e89f1f0ddc84480ced",
    "cross_cats_sorted": [
      "math.CO",
      "stat.TH"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.ST",
    "submitted_at": "2026-05-15T02:02:36Z",
    "title_canon_sha256": "d84c2848a73d8cfd66c1a6b9dc5249c9d1964bf7bbc5e217f37f93f69b68e362"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15531",
    "kind": "arxiv",
    "version": 1
  }
}