Pith Number

pith:HIZMU5UL

pith:2026:HIZMU5ULBLZB5JVSPSTS4ODD7A

not attested not anchored not stored refs resolved

A Unified Geometric Framework for Weighted Contrastive Learning

Benoit Dufumier, Edouard Duchesnay, Raphael Vock

Weighted InfoNCE objectives correspond to distance geometry problems whose solutions fix the geometry of optimal embeddings.

arxiv:2605.13943 v1 · 2026-05-13 · cs.LG

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{HIZMU5ULBLZB5JVSPSTS4ODD7A}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

weighted InfoNCE objectives can be interpreted as Distance Geometry Problems, where the weighting scheme specifies the target geometry to be realized by the representation. This viewpoint yields exact characterizations of the optimal embeddings for several supervised and weakly supervised objectives.

C2weakest assumption

That the contrastive objective can be exactly recast as a distance geometry problem whose solution is attained by gradient descent on the embedding parameters, without additional constraints from finite batch sizes, temperature scaling, or optimization dynamics.

C3one line summary

Weighted InfoNCE objectives realize specific target geometries in embedding space, with SupCon producing size-dependent inter-class similarities under imbalance while Soft SupCon and certain continuous variants preserve regular simplex or unique optima.

References

51 extracted · 51 resolved · 3 Pith anchors

[1] International Conference on Machine Learning , pages= 2021

[2] Journal of Mathematical Psychology , volume= 1981

[3] Cognitive psychology , volume= 1975

[4] Advances in Neural Information Processing Systems , volume=

[5] arXiv preprint arXiv:2407.18134 , year=

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:13.831100Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3a32ca768b0af21ea6b27ca72e3863f839133dc8478f30ea8e1b34b7eb39b188

Aliases

arxiv: 2605.13943 · arxiv_version: 2605.13943v1 · doi: 10.48550/arxiv.2605.13943 · pith_short_12: HIZMU5ULBLZB · pith_short_16: HIZMU5ULBLZB5JVS · pith_short_8: HIZMU5UL

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/HIZMU5ULBLZB5JVSPSTS4ODD7A \
  | 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: 3a32ca768b0af21ea6b27ca72e3863f839133dc8478f30ea8e1b34b7eb39b188

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a499e9897a23916596d8abf70f52558e333a6cc92b72e0c7cbbe71dea9cb09c5",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-13T17:48:36Z",
    "title_canon_sha256": "4b3ccdb39002a04cbbba6ea41f1b392a68938f58b3b1981e9a5acdd5f2bbfe5b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13943",
    "kind": "arxiv",
    "version": 1
  }
}