Paper Citation Record · LEDGER

Understanding Learning Invariance in Deep Linear Networks

As of 17 August 2026, this Paper Citation Record lists 38 of 38 outbound references and 3 inbound Pith citation observations for arXiv:2506.13714.

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

pith.paper-citation-record.v1
2506.13714 v1

Coverage vector

measured 38 of 38 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-15T20:08:54.979685Z

measured 41 of 41 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-16T06:30:59.297886+00:00

measured 3 of 3 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-15T14:39:45.652256Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-07-04T15:09:55.451629Z

Reference resolution

38 of 38 outbound references displayed

verified exact3
verified fuzzy20
unresolved14
parse uncertain1
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation b0d4df81-1f56-4b38-87c0-e3bcb148e45e · outbound

This paper cites an unresolved cited work.

Understanding Learning Invariance in Deep Linear Networks Unresolved cited work

Reference 1

Resolution: parse uncertain; raw_fallback, observed 2026-08-15T20:08:56.448816Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 0c5f1aca-c08e-4bd9-b785-f076a4f046a4 · outbound

This paper cites That is to say,G is a projection operator fromX to the subspace allG-fixed points.

Understanding Learning Invariance in Deep Linear Networks That is to say,G is a projection operator fromX to the subspace allG-fixed points

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.434320Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 81834d9c-c11c-46fe-96b6-23fcb3d0b98f · outbound

This paper cites an unresolved cited work.

Understanding Learning Invariance in Deep Linear Networks Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-08-15T20:08:56.418158Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 74dc89b0-d7c2-49c5-96ba-1add6f4f4ef3 · outbound

This paper cites Emergent Equivariance in Deep Ensembles.

Understanding Learning Invariance in Deep Linear Networks Emergent Equivariance in Deep Ensembles

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.213961Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d314355b-a749-4109-9a35-52f2a1bb62c9 · outbound

This paper cites URL https://link.springer.com/10.1007/ 978-3-662-07931-7.

Understanding Learning Invariance in Deep Linear Networks URL https://link.springer.com/10.1007/ 978-3-662-07931-7

Reference 7

Resolution: verified exact; doi, observed 2026-08-15T20:08:55.140858Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation cc6e186e-eda9-468f-8cd3-33757972807c · outbound

This paper cites URL https:// doi.org/10.1007/978-1-4757-2189-8_9.

Understanding Learning Invariance in Deep Linear Networks URL https:// doi.org/10.1007/978-1-4757-2189-8_9

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.336021Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9c0f8dac-de90-45fc-828e-dfe785a965d3 · outbound

This paper cites Auto-Encoding Variational Bayes.

Understanding Learning Invariance in Deep Linear Networks Auto-Encoding Variational Bayes

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.446468Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation dccc80f9-214c-4526-8dc4-4bb60cd9035e · outbound

This paper cites eP11 eP12 eP† 12 eP22 #2.

Understanding Learning Invariance in Deep Linear Networks eP11 eP12 eP† 12 eP22 #2

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.169944Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 68ec75aa-f05c-4af8-bb3b-d988af725372 · outbound

This paper cites Enhanced Convolutional Neural Tangent Kernels.

Understanding Learning Invariance in Deep Linear Networks Enhanced Convolutional Neural Tangent Kernels

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.457564Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1dbb341f-3d6e-4029-80ef-fb3dd1db2aed · outbound

This paper cites cc/paper_files/paper/2019/file/ bdbca288fee7f92f2bfa9f7012727740-Paper.

Understanding Learning Invariance in Deep Linear Networks cc/paper_files/paper/2019/file/ bdbca288fee7f92f2bfa9f7012727740-Paper

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.711425Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 39083538-1a25-4f1c-880c-8c0474d0e191 · outbound

This paper cites Ruben, G.

Understanding Learning Invariance in Deep Linear Networks Ruben, G

Reference 17

Resolution: verified exact; raw_fallback, observed 2026-08-15T20:08:55.538386Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation dab9a8ab-b38c-4cd7-93c1-5b12073dbf2a · outbound

This paper cites Soltanolkotabi, M., Javanmard, A., and Lee, J.

Understanding Learning Invariance in Deep Linear Networks Soltanolkotabi, M., Javanmard, A., and Lee, J

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.644271Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 8a8f1083-a41c-4529-809a-d3edfffdfad8 · outbound

This paper cites Song, Z., Woodruff, D.

Understanding Learning Invariance in Deep Linear Networks Song, Z., Woodruff, D

Reference 19

Resolution: verified exact; raw_fallback, observed 2026-08-15T20:08:55.369352Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation aa9f5e86-7dc8-4452-8096-711a5b0b6135 · outbound

This paper cites URL https://epubs.siam.org/doi/abs/10.

Understanding Learning Invariance in Deep Linear Networks URL https://epubs.siam.org/doi/abs/10

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.673021Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bb274e56-e99d-4422-a7db-0192a15a50c1 · outbound

This paper cites cc/paper_files/paper/2017/file/ f22e4747da1aa27e363d86d40ff442fe-Paper.

Understanding Learning Invariance in Deep Linear Networks cc/paper_files/paper/2017/file/ f22e4747da1aa27e363d86d40ff442fe-Paper

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.763025Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 40c6668b-eb48-474f-8d5d-0138e661b463 · outbound

This paper cites An Introduction to Electrocatalyst Design using Machine Learning for Renewable Energy Storage.

Understanding Learning Invariance in Deep Linear Networks An Introduction to Electrocatalyst Design using Machine Learning for Renewable Energy Storage

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.768319Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e4d1b084-c33a-4101-833c-85c9ce117b77 · outbound

This paper cites We have GρX (h) = Z G ρX (g)dλ(g)ρX (h) = Z G ρX (gh)dλ(g) = Z G ρX (gh)dλ(gh) =G.

Understanding Learning Invariance in Deep Linear Networks We have GρX (h) = Z G ρX (g)dλ(g)ρX (h) = Z G ρX (gh)dλ(g) = Z G ρX (gh)dλ(gh) =G

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.334328Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 086b4f34-4890-46ee-8349-078263d9817d · outbound

This paper cites (37) 17 Towards a Theoretical Understanding of Learning Invariance in Deep Linear Networks via Loss Landscapes.

Understanding Learning Invariance in Deep Linear Networks (37) 17 Towards a Theoretical Understanding of Learning Invariance in Deep Linear Networks via Loss Landscapes

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.291169Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 15b72faa-2262-4894-a930-d248e567c972 · outbound

This paper cites eX1:deX† 1:d −1 0d,d0−d 0d0−d,d 0d0−d,d0−d # = eP−1 Id0− eP−1(Λg− Id0) eP−1(Λg− Id0) + eP−1. (54) We can see that eP 2 eP−2−.

Understanding Learning Invariance in Deep Linear Networks eX1:deX† 1:d −1 0d,d0−d 0d0−d,d 0d0−d,d0−d # = eP−1 Id0− eP−1(Λg− Id0) eP−1(Λg− Id0) + eP−1. (54) We can see that eP 2 eP−2−

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.239283Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 1e5f2f8e-94fc-4490-9e75-e8d20526f977 · outbound

This paper cites The theorem is proved.

Understanding Learning Invariance in Deep Linear Networks The theorem is proved

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.153361Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation f486ed96-96ee-478e-8a24-3aff91e58abd · outbound

This paper cites They are all in the form of U inv Σinv I V invT P−1, whereI∈ [d]r.

Understanding Learning Invariance in Deep Linear Networks They are all in the form of U inv Σinv I V invT P−1, whereI∈ [d]r

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.055754Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 3f0de3c0-6cc7-4b50-9970-d9464c007166 · outbound

This paper cites They are all in the form of U da Σda I V daT Q−1, whereI∈ [d]r.

Understanding Learning Invariance in Deep Linear Networks They are all in the form of U da Σda I V daT Q−1, whereI∈ [d]r

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.037935Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 8ca3facb-fc85-4450-8816-8d790602a728 · outbound

This paper cites They are all in the form of U reg Σreg I V regT B(λ)−1P−1, whereI∈ [m]r.

Understanding Learning Invariance in Deep Linear Networks They are all in the form of U reg Σreg I V regT B(λ)−1P−1, whereI∈ [m]r

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:55.961838Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation d6a0e3e8-9005-44f9-a2da-22cb7e8cd42e · outbound

This paper cites This suggests that it is more difficult to learn invariance from the data when the model has more parameters.

Understanding Learning Invariance in Deep Linear Networks This suggests that it is more difficult to learn invariance from the data when the model has more parameters

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:55.887640Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 1584c763-8427-4d63-8200-426c7d74085b · outbound

This paper cites For underdetermined linear models, i.e., when the number of data points exceeds the input dimension, Proposition 3.10 shows that all critical points are invariant.

Understanding Learning Invariance in Deep Linear Networks For underdetermined linear models, i.e., when the number of data points exceeds the input dimension, Proposition 3.10 shows that all critical points are invariant

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:55.816110Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation cc64a122-830a-4b4f-a79b-2510ad42941b · outbound

This paper cites This suggests that the invariance learned from the data is fairly robust.

Understanding Learning Invariance in Deep Linear Networks This suggests that the invariance learned from the data is fairly robust

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:55.733365Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 82eab2dd-dec3-4753-955d-5ec57288a5a4 · outbound

This paper cites an unresolved cited work.

Understanding Learning Invariance in Deep Linear Networks Unresolved cited work

Reference 1987

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.279121Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a10fb092-cc03-4150-a59d-280119ae28e1 · outbound

This paper cites Gunasekar, S., Lee, J.

Understanding Learning Invariance in Deep Linear Networks Gunasekar, S., Lee, J

Reference 1992

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.883045Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 05da31be-4616-48bc-a32f-feb3d899b00d · outbound

This paper cites LoRA: Low-Rank Adaptation of Large Language Models.

Understanding Learning Invariance in Deep Linear Networks LoRA: Low-Rank Adaptation of Large Language Models

Reference 2017

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.369933Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b8e4729e-e0db-4e74-a8e2-b06f419a9a97 · outbound

This paper cites cc/paper_files/paper/2018/file/ 0e98aeeb54acf612b9eb4e48a269814c-Paper.

Understanding Learning Invariance in Deep Linear Networks cc/paper_files/paper/2018/file/ 0e98aeeb54acf612b9eb4e48a269814c-Paper

Reference 2018

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.868551Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 3d984735-887d-4a0c-8cd4-c3e4c701d2bd · outbound

This paper cites Mei, S., Misiakiewicz, T., and Montanari, A.

Understanding Learning Invariance in Deep Linear Networks Mei, S., Misiakiewicz, T., and Montanari, A

Reference 2019

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.764017Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 50ba2d76-d339-48e1-b817-15220a76ad03 · outbound

This paper cites Xu, Z., Min, H., Tarmoun, S., Mallada, E., and Vidal, R.

Understanding Learning Invariance in Deep Linear Networks Xu, Z., Min, H., Tarmoun, S., Mallada, E., and Vidal, R

Reference 2020

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.559980Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation c9642859-4417-44b7-ac01-d659d3610142 · outbound

This paper cites Optimization Dynamics of Equivariant and Augmented Neural Networks.

Understanding Learning Invariance in Deep Linear Networks Optimization Dynamics of Equivariant and Augmented Neural Networks

Reference 2021

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.558665Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 27c72660-2e4b-42ef-928e-4728681de4d2 · outbound

This paper cites an unresolved cited work.

Understanding Learning Invariance in Deep Linear Networks Unresolved cited work

Reference 2023

Resolution: unresolved; raw_fallback, observed 2026-08-15T20:08:56.964127Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation af723b18-3f7b-46e0-9404-14b091d2dc59 · outbound

This paper cites URL https://link.springer.com/10.1007/ s10107-024-02058-3.

Understanding Learning Invariance in Deep Linear Networks URL https://link.springer.com/10.1007/ s10107-024-02058-3

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.452640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 99c08639-ab47-4159-840d-38d8a16e7dfa · outbound

This paper cites URL https://proceedings.

Understanding Learning Invariance in Deep Linear Networks URL https://proceedings

Reference 2284

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.543678Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 01de1cb9-7bb9-4148-85fe-916a2fd6f164 · outbound

This paper cites e3nn: Euclidean Neural Networks.

Understanding Learning Invariance in Deep Linear Networks e3nn: Euclidean Neural Networks

Reference 5454

Resolution: unresolved; no resolver link, observed 2026-08-15T20:08:54.209082Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a10d0461-42b2-48e6-8312-c2660b5996d4 · outbound

This paper cites URL https: //openreview.net/forum?id=DnVjDRLwVu.

Understanding Learning Invariance in Deep Linear Networks URL https: //openreview.net/forum?id=DnVjDRLwVu

Reference 6155

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T20:08:56.898098Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Pith citing papers

Observation 32f1b361-62b7-4faa-b466-3fa4f3d5a73a · inbound

Conservation Laws from Data Symmetry in Neural Networks cites this paper.

Conservation Laws from Data Symmetry in Neural Networks Understanding Learning Invariance in Deep Linear Networks

Reference 19

Resolution: metadata mismatch; arxiv_id, observed 2026-07-03T04:27:36.903387Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 6fca3369-70e8-48c5-9d2e-77316f11b03e · inbound

Equivariance and Augmentation for Bayesian Neural Networks cites this paper.

Equivariance and Augmentation for Bayesian Neural Networks Understanding Learning Invariance in Deep Linear Networks

Reference 19

Resolution: verified exact; arxiv_id, observed 2026-07-04T15:09:55.453322Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation fa90a9f4-9abd-4e91-b010-aa3394eb7fec · inbound

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks cites this paper.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Understanding Learning Invariance in Deep Linear Networks

Reference 39

Resolution: unresolved; no resolver link, observed 2026-08-15T14:39:45.652256Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.