Pith Number

pith:NQPQJSB7

pith:2026:NQPQJSB76UDZVURRCBGP4FM6EE

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Stochastic Attention via Langevin Dynamics on the Modern Hopfield Energy

Abdulrahman Alswaidan, Jeffrey D. Varner

Attention retrieval equals one gradient step on the modern Hopfield energy, so Langevin dynamics yields a training-free stochastic sampler governed by temperature.

arxiv:2603.06875 v3 · 2026-03-06 · cs.LG · q-fin.CP

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Claims

C1strongest claim

We showed that this computation is one step of gradient descent on the modern Hopfield energy, and that Langevin sampling from the corresponding Boltzmann distribution yielded stochastic attention, a training-free sampler controlled by a single temperature parameter.

C2weakest assumption

The assumption that the energy gradient exactly equals the attention map, allowing direct application of Langevin dynamics to produce valid samples without further modeling or approximations.

C3one line summary

Langevin sampling on the modern Hopfield energy produces training-free stochastic attention that transitions from exact retrieval to generation as temperature rises, with an entropy inflection condition marking the shift.

References

40 extracted · 40 resolved · 1 Pith anchors

[1] Gomez, Łukasz Kaiser, and Illia Polosukhin 2017

[2] Proceedings of the National Academy of Sci- ences79(8), 2554–2558 (Apr 1982) 1982 · doi:10.1073/pnas.79.8.2554

[3] Dmitry Krotov and John J. Hopfield. Dense associative memory for pattern recognition. In Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016 2016

[4] Large associative memory problem in neurobiology and machine learning 2021

[5] Hopfield networks is all you need 2021

Receipt and verification

First computed	2026-05-17T23:38:59.735476Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6c1f04c83ff5079ad231104cfe159e212eca9c6476cdecd19257a51b07c5b025

Aliases

arxiv: 2603.06875 · arxiv_version: 2603.06875v3 · doi: 10.48550/arxiv.2603.06875 · pith_short_12: NQPQJSB76UDZ · pith_short_16: NQPQJSB76UDZVURR · pith_short_8: NQPQJSB7

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/NQPQJSB76UDZVURRCBGP4FM6EE \
  | 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: 6c1f04c83ff5079ad231104cfe159e212eca9c6476cdecd19257a51b07c5b025

Canonical record JSON

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    "abstract_canon_sha256": "61bbb0501458eb3bb092e59a5ca21b49717f83eb314e64c7afcd4358bc03ef88",
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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-03-06T20:50:30Z",
    "title_canon_sha256": "5cc7d70025a2d57d48439e22dc7af2837180328af88442e501d52e0b6b0df148"
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