Pith Number

pith:YTFEG2WT

pith:2026:YTFEG2WTBV6SNMDQIESPCMGTD7

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Generalizing Score-based generative models for Heavy-tailed Distributions

Gabriel Cardoso, Sylvan Le Corff, Thomas Romary, Tiziano Fassina

Early stopping plus normalizing flow initialization extends score-based models to any heavy-tailed target with KL convergence.

arxiv:2603.00772 v2 · 2026-02-28 · stat.ML · cs.LG

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Claims

C1strongest claim

Combining early stopping with a suitable initialization is sufficient to extend the diffusion framework to any target distribution; we establish the well-posedness of the backward process and prove convergence of the approximated diffusion in KL divergence. Novel theoretical guarantees for generation with normalizing flows hold under mild conditions on the flow family and without any assumption on the tail behavior of the target distribution.

C2weakest assumption

The normalizing flow must be expressive enough to capture the tail behavior of the target distribution so that it provides a useful initialization prior for the subsequent SGM refinement step.

C3one line summary

Early stopping plus normalizing flow initialization extends diffusion models to any target distribution with proven KL convergence, and a hybrid flow-SGM pipeline captures heavy tails without tail-specific assumptions.

References

15 extracted · 15 resolved · 1 Pith anchors

[1] Imagen 3.arXiv preprint arXiv:2408.07009, 2024 2022

[2] Adam: A Method for Stochastic Optimization 2024 · arXiv:1412.6980

[3] dµX0(x0) = Z CT ×Rd Z T 0 ∥At(x)−a t(x)∥2 2b 2 t dtdµ X,X0(x, x0) = Z CT Z T 0 ∥At(x)−a t(x)∥2 2b 2 t dtdµ X(x) =E " 1 2 Z T 0 1 b2s ∥As(X)−a s(X)∥2 ds # ,(22) which concludes the proof. A.2. Lemmas r 2003

[4] Using the general property ∆f(x) f(x) = ∆f(x) +∥∇f(x)∥ 2 , we can write ∂t log ˜pT−t (x)ρ(x) + ¯αt h ρ(x)∇log ˜pT−t (x)·x+∇ρ(x)·x+d ρ(x) i + ¯g2 t 2 h ρ(x) [∆ log ˜pT−t (x) +∥∇log ˜pT−t (x)∥2] + ∆ρ(x) 2025

[5] To obtain Equation (12), using the definition ofY t write N−1X k=0 Z tk+1 tk ¯g2 t E ∥∇log⃗ pT−t k(← −X tk)− ∇log⃗ pT−t (← −X t)∥2 dt= N−1X k=0 Z tk+1 tk ¯g2 t E ∥Ytk −Y t∥2 dt 2018

Formal links

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Cited by

1 paper in Pith

Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training

Receipt and verification

First computed	2026-05-17T23:39:04.391761Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c4ca436ad30d7d26b0704124f130d31fc5c8097f93bcc8c9ea2551af32b06fd8

Aliases

arxiv: 2603.00772 · arxiv_version: 2603.00772v2 · doi: 10.48550/arxiv.2603.00772 · pith_short_12: YTFEG2WTBV6S · pith_short_16: YTFEG2WTBV6SNMDQ · pith_short_8: YTFEG2WT

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YTFEG2WTBV6SNMDQIESPCMGTD7 \
  | 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: c4ca436ad30d7d26b0704124f130d31fc5c8097f93bcc8c9ea2551af32b06fd8

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6cf614ea558a43622709a674a8cb1d2b212b39f8796336cd94773189d5868608",
    "cross_cats_sorted": [
      "cs.LG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-02-28T18:37:10Z",
    "title_canon_sha256": "c1345984924013cf27d2b56e69e2eec01b2aa5755928db1180f4d57113cb21b2"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.00772",
    "kind": "arxiv",
    "version": 2
  }
}