Paper Citation Record · LEDGER

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation

As of 20 August 2026, this Paper Citation Record lists 21 of 21 outbound references and 4 inbound Pith citation observations for arXiv:2501.18530.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2501.18530 v3

Coverage vector

measured 21 of 21 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-09T23:13:39.385811Z

measured 25 of 25 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-19T06:32:44.657259+00:00

measured 4 of 4 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-15T23:53:45.280757Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-07-01T09:55:40.811765Z

Reference resolution

21 of 21 outbound references displayed

verified exact5
verified fuzzy4
unresolved7
parse uncertain0
malformed identifier5
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 66485700-259e-4168-92b4-8f4b82a517ad · outbound

This paper cites an unresolved cited work.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-09T23:13:41.288753Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 89a76c59-7e8f-4cbb-8a69-3eac7d8d2113 · outbound

This paper cites Notice that we have re-normalised Sa 2 by √ d in order to work with matrices with O(1) eigenvalues.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Notice that we have re-normalised Sa 2 by √ d in order to work with matrices with O(1) eigenvalues

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T23:13:41.523917Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 5079e297-eaf5-420b-bf0f-4d93d13bd10b · outbound

This paper cites Pacelli, R., Ariosto, S., Pastore, M., Ginelli, F., Gher- ardi, M., and Rotondo, P.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Pacelli, R., Ariosto, S., Pastore, M., Ginelli, F., Gher- ardi, M., and Rotondo, P

Reference 6

Resolution: verified exact; arxiv_id_nonexistent, observed 2026-08-09T23:13:39.866455Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation dbd7e3f3-351b-45c7-8929-4f7dce997bb0 · outbound

This paper cites URL https://doi.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation URL https://doi

Reference 7

Resolution: verified exact; arxiv_id_nonexistent, observed 2026-08-09T23:13:40.588875Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 9c9718fd-573e-4c4a-987a-ab4fe65a27a0 · outbound

This paper cites cc/paper_files/paper/2020/file/ 9b8b50fb590c590ffbf1295ce92258dc-Paper.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation cc/paper_files/paper/2020/file/ 9b8b50fb590c590ffbf1295ce92258dc-Paper

Reference 14

Resolution: verified exact; arxiv_id_nonexistent, observed 2026-08-09T23:13:39.702098Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 6c93cc29-4c25-4988-8613-480b2b84b978 · outbound

This paper cites Formally the integration is over the imaginary axis iR.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Formally the integration is over the imaginary axis iR

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T23:13:41.380604Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 3ba32464-9261-4a17-b21a-5764e3d351d4 · outbound

This paper cites an unresolved cited work.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-09T23:13:41.319081Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 97a41d99-486a-4d4c-93ac-68d4ddbb51bd · outbound

This paper cites (84) Here I(¯S0 2; Y(ˆq2)) is the MI related to the channel (82), µY(ˆq2) is the asymptotic spectral law of the observation matrix Y(ˆq2).

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation (84) Here I(¯S0 2; Y(ˆq2)) is the MI related to the channel (82), µY(ˆq2) is the asymptotic spectral law of the observation matrix Y(ˆq2)

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T23:13:41.303888Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation d698a9c4-4e15-4eec-b378-76e699280ac9 · outbound

This paper cites Aiudi, R., Pacelli, R., Baglioni, P., Vezzani, A., Burioni, R., and Rotondo, P.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Aiudi, R., Pacelli, R., Baglioni, P., Vezzani, A., Burioni, R., and Rotondo, P

Reference 39

Resolution: malformed identifier; no resolver link, observed 2026-08-09T23:13:38.747853Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1ad51555-d709-4234-99d6-7ba2189bd757 · outbound

This paper cites com/doi/abs/10.1142/0271.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation com/doi/abs/10.1142/0271

Reference 271

Resolution: malformed identifier; no resolver link, observed 2026-08-09T23:13:38.835285Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 871352b4-5cb9-407b-a17a-f3d42c064a16 · outbound

This paper cites doi: 10.1063/1.524438.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation doi: 10.1063/1.524438

Reference 1980

Resolution: verified exact; arxiv_id_nonexistent, observed 2026-08-09T23:13:40.308256Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 9f790c6e-7333-4ad2-ae6d-6a050c9b94c3 · outbound

This paper cites doi: 10.1007/ 978-1-4612-0745-0 2.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation doi: 10.1007/ 978-1-4612-0745-0 2

Reference 1996

Resolution: unresolved; no resolver link, observed 2026-08-09T23:13:38.844649Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c76f0da7-ae5c-463a-a32c-76df44404261 · outbound

This paper cites URL https: //doi.org/10.1088/0305-4470/31/38/012.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation URL https: //doi.org/10.1088/0305-4470/31/38/012

Reference 1998

Resolution: unresolved; no resolver link, observed 2026-08-09T23:13:39.018381Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 120d4fd4-a583-43ed-afc3-8b32cf2bdbf8 · outbound

This paper cites cc/paper_files/paper/2018/file/ 84f0f20482cde7e5eacaf7364a643d33-Paper.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation cc/paper_files/paper/2018/file/ 84f0f20482cde7e5eacaf7364a643d33-Paper

Reference 2018

Resolution: malformed identifier; no resolver link, observed 2026-08-09T23:13:38.795262Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b454a136-9400-4d2e-8ca4-125bbdb34a6c · outbound

This paper cites Fundamental limits of overparametrized shallow neural networks for supervised learning.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Fundamental limits of overparametrized shallow neural networks for supervised learning

Reference 2020

Resolution: unresolved; no resolver link, observed 2026-08-09T23:13:38.806469Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f9a2e7a3-9fef-4618-a4eb-389d1b7f660d · outbound

This paper cites cc/paper_files/paper/2021/file/ b24d21019de5e59da180f1661904f49a-Paper.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation cc/paper_files/paper/2021/file/ b24d21019de5e59da180f1661904f49a-Paper

Reference 2021

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T23:13:41.558608Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 82ad23a8-2649-491b-8aef-470cd673c154 · outbound

This paper cites Guionnet, A.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Guionnet, A

Reference 2022

Resolution: verified exact; arxiv_id_nonexistent, observed 2026-08-09T23:13:40.873043Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation de444104-3403-4867-b6a5-7932e370442d · outbound

This paper cites Del Giudice, P., Franz, S., and Virasoro, M.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Del Giudice, P., Franz, S., and Virasoro, M

Reference 2023

Resolution: malformed identifier; arxiv_id_nonexistent, observed 2026-08-09T23:13:41.080186Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 1ed06b74-037c-47c9-96b3-aedf7e3fd3aa · outbound

This paper cites an unresolved cited work.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Unresolved cited work

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-09T23:13:38.830264Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a14938c3-eea0-43ab-bb14-0d2f1ff937f8 · outbound

This paper cites Coding schemes in neural networks learning classification tasks.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation Coding schemes in neural networks learning classification tasks

Reference 2025

Resolution: malformed identifier; no resolver link, observed 2026-08-09T23:13:38.928818Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fdfbf3ac-b68f-420e-9eb5-5a098637f457 · outbound

This paper cites On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance.

Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance

Reference 8772

Resolution: unresolved; no resolver link, observed 2026-08-09T23:13:38.800507Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation 08e55951-6378-4f96-aafb-cd416c0f2510 · inbound

Information-theoretic reduction of deep neural networks to linear models in the overparametrized proportional regime cites this paper.

Information-theoretic reduction of deep neural networks to linear models in the overparametrized proportional regime Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-15T23:53:45.280757Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f1cd2bcc-79ba-4acf-be0e-c9c36e7a0874 · inbound

Microscopic and collective signatures of feature learning in neural networks cites this paper.

Microscopic and collective signatures of feature learning in neural networks Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation

Reference 52

Resolution: unresolved; no resolver link, observed 2026-08-05T14:46:58.450056Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0cb58844-94df-4d22-a4f3-0cea5a276554 · inbound

Kernel Renormalization in Bayesian Deep Neural Networks: the Equivalent Wishart Ansatz in the Proportional Regime cites this paper.

Kernel Renormalization in Bayesian Deep Neural Networks: the Equivalent Wishart Ansatz in the Proportional Regime Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation

Reference 72

Resolution: verified exact; arxiv_id, observed 2026-07-27T01:19:45.486658Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 3cbd02fb-0489-42c3-b393-78e5c295a582 · inbound

Explaining Machine Learning and Memorization with Statistical Mechanics cites this paper.

Explaining Machine Learning and Memorization with Statistical Mechanics Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation

Reference 22

Resolution: verified exact; arxiv_id, observed 2026-07-27T01:19:45.486658Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.