Pith Number

pith:4KXYZZYO

pith:2026:4KXYZZYOAL6CTQOAHW6BLN3YHB

not attested not anchored not stored refs pending

Statistical Consistency and Generalization of Contrastive Representation Learning

Tianbao Yang, Xiyuan Wei, Yiming Ying, Yuanfan Li

Contrastive loss is statistically consistent with optimal ranking in retrieval tasks

arxiv:2605.02116 v2 · 2026-05-04 · cs.LG

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\pithnumber{4KXYZZYOAL6CTQOAHW6BLN3YHB}

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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

We show that the contrastive loss is statistically consistent with optimal ranking and derive generalization bounds of order O(1/m + 1/sqrt(n)) and O(1/sqrt(m) + 1/sqrt(n)) for supervised and self-supervised CRL respectively.

C2weakest assumption

The paper does not state the weakest assumption explicitly in the abstract; the bounds and consistency results are likely to rest on standard statistical-learning assumptions such as i.i.d. sampling, bounded loss functions, and appropriate regularity conditions on the data distribution that are not detailed here.

C3one line summary

Contrastive representation learning is statistically consistent for optimal retrieval and admits generalization bounds of order O(1/m + 1/sqrt(n)) supervised and O(1/sqrt(m) + 1/sqrt(n)) self-supervised that benefit from large negative-sample counts m.

Receipt and verification

First computed	2026-05-21T01:04:26.657450Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e2af8ce70e02fc29c1c03dbc15b7783859a0af203aa9e0e93f02205a031e5777

Aliases

arxiv: 2605.02116 · arxiv_version: 2605.02116v2 · doi: 10.48550/arxiv.2605.02116 · pith_short_12: 4KXYZZYOAL6C · pith_short_16: 4KXYZZYOAL6CTQOA · pith_short_8: 4KXYZZYO

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/4KXYZZYOAL6CTQOAHW6BLN3YHB \
  | 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: e2af8ce70e02fc29c1c03dbc15b7783859a0af203aa9e0e93f02205a031e5777

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9800b4b057f16bb7270417674de90b5c0c839a1d45f18e8ce12d401ee95a77fc",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-04T00:38:29Z",
    "title_canon_sha256": "e31927de8b7349f6052390c6f4c1970324cac524aeb4393aaf41facb000fe3f1"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.02116",
    "kind": "arxiv",
    "version": 2
  }
}