Pith Number

pith:6CRN3T5H

pith:2026:6CRN3T5HCWOUVF6KEA4FGDTLS7

not attested not anchored not stored refs pending

Spherical Flows for Sampling Categorical Data

Gabriele Steidl, Gregor Kornhardt, Jannis Chemseddine

Spherical flows using the von Mises-Fisher distribution reduce categorical sequence sampling to solving a scalar ODE in cosine similarity.

arxiv:2605.05629 v3 · 2026-05-07 · stat.ML · cs.CL · cs.LG

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Claims

C1strongest claim

Exploiting the radial symmetry of the vMF density we reduce the continuity equation on S^{d-1} to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on (S^{d-1})^L both decompose into posterior-weighted tangent sums.

C2weakest assumption

That the learned posterior (trained only by cross-entropy) is sufficiently accurate to produce stable posterior-weighted sums for both velocity and score during sampling on real discrete data.

C3one line summary

Spherical vMF flows reduce the continuity equation on the sphere to a scalar ODE in cosine similarity, enabling posterior-weighted sampling of categorical sequences via cross-entropy trained posteriors.

Formal links

2 machine-checked theorem links

Cited by

1 paper in Pith

Self-conditioned Flow Map Language Models via Fixed-point Flows

Receipt and verification

First computed	2026-06-03T01:05:50.993923Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f0a2ddcfa7159d4a97ca2038530e6b97dbd71545b389f9b26bee647b6fec00fc

Aliases

arxiv: 2605.05629 · arxiv_version: 2605.05629v3 · doi: 10.48550/arxiv.2605.05629 · pith_short_12: 6CRN3T5HCWOU · pith_short_16: 6CRN3T5HCWOUVF6K · pith_short_8: 6CRN3T5H

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/6CRN3T5HCWOUVF6KEA4FGDTLS7 \
  | 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: f0a2ddcfa7159d4a97ca2038530e6b97dbd71545b389f9b26bee647b6fec00fc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3576557003e26bc940a50eba6e5d6d3e4b85351aa2a2f508f5f4a974aa947145",
    "cross_cats_sorted": [
      "cs.CL",
      "cs.LG"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-05-07T03:34:00Z",
    "title_canon_sha256": "0a85d3ba640e33adae517a0da42237e37a0d8b61a3f63c2e9037dd15e9e59ac0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.05629",
    "kind": "arxiv",
    "version": 3
  }
}