Paper Citation Record · LEDGER

Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

As of 8 August 2026, this Paper Citation Record lists 0 of 0 outbound references and 17 inbound Pith citation observations for arXiv:2106.15933.

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

pith.paper-citation-record.v1
2106.15933 v2

Coverage vector

measured 0 of 0 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links

measured 17 of 17 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-08T06:32:00.761636+00:00

measured 17 of 17 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-08T19:58:26.454460Z

measured 0 of 1 external citation measurements

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

Source: pith, observed 2026-07-10T16:07:20.384070Z

Reference resolution

0 of 0 outbound references displayed

verified exact0
verified fuzzy0
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External citation measurements

No source-named external measurement is stored.

Outbound references

No outbound reference observations are available for this paper version.

Pith citing papers

Observation c5dc415e-0547-4423-bd7c-ffb2aceb4d9d · inbound

Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs cites this paper.

Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 18

Resolution: metadata mismatch; arxiv_id, observed 2026-05-24T11:29:24.943153Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation f0835da8-74a9-4d74-b960-dd28cc4b84d8 · inbound

Parameter Symmetry Potentially Unifies Deep Learning Theory cites this paper.

Parameter Symmetry Potentially Unifies Deep Learning Theory Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 33

Resolution: unresolved; no resolver link, observed 2026-08-08T19:58:26.454460Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 769afe19-6e56-443e-aa3b-48ef611a577f · inbound

Adaptive kernel predictors from feature-learning infinite limits of neural networks cites this paper.

Adaptive kernel predictors from feature-learning infinite limits of neural networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-08T11:19:06.913318Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3a8f20e4-0c1c-4f62-8650-bbd5be08d4ff · inbound

Gradient Flow Equations for Deep Linear Neural Networks: A Survey from a Network Perspective cites this paper.

Gradient Flow Equations for Deep Linear Neural Networks: A Survey from a Network Perspective Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-03T22:31:44.546962Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 586fda19-a8a8-40da-b22c-a0822fc908b7 · inbound

Over-Alignment vs Over-Fitting: The Role of Feature Learning Strength in Generalization cites this paper.

Over-Alignment vs Over-Fitting: The Role of Feature Learning Strength in Generalization Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 2018

Resolution: unresolved; no resolver link, observed 2026-08-03T06:00:14.652992Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9b281b65-6c0a-49ee-bbb4-85a0350aab2a · inbound

There Will Be a Scientific Theory of Deep Learning cites this paper.

There Will Be a Scientific Theory of Deep Learning Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 139

Resolution: verified exact; arxiv_id, observed 2026-05-11T15:21:09.144024Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 4255e4da-dfcd-46a1-be07-8dca82121aa8 · inbound

Low-Rank Adaptation Redux for Large Models cites this paper.

Low-Rank Adaptation Redux for Large Models Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 76

Resolution: metadata mismatch; arxiv_id, observed 2026-05-11T14:26:03.923173Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation e986a627-14a1-4570-890d-91fdda2ab573 · inbound

A Theory of Saddle Escape in Deep Nonlinear Networks cites this paper.

A Theory of Saddle Escape in Deep Nonlinear Networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 23

Resolution: verified exact; arxiv_id, observed 2026-05-11T16:51:05.621784Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 0580cf1d-33f6-493e-a206-c102a1e917a9 · inbound

A Theory of Saddle Escape in Deep Nonlinear Networks cites this paper.

A Theory of Saddle Escape in Deep Nonlinear Networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 23

Resolution: verified exact; arxiv_id, observed 2026-05-11T02:25:54.524070Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation cc13ca84-3b89-473a-b861-b4f28adf5228 · inbound

A Theory of Saddle Escape in Deep Nonlinear Networks cites this paper.

A Theory of Saddle Escape in Deep Nonlinear Networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 23

Resolution: verified exact; arxiv_id, observed 2026-07-01T00:45:12.010776Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 2c200eaa-fce0-4c85-9201-bbd28c6cc082 · inbound

The Implicit Bias of Depth: From Neural Collapse to Softmax Codes cites this paper.

The Implicit Bias of Depth: From Neural Collapse to Softmax Codes Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 111

Resolution: verified exact; arxiv_id, observed 2026-05-25T05:26:38.967729Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 1ef28b9a-a2f5-49ec-b042-a50300625c0b · inbound

Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention cites this paper.

Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 59

Resolution: metadata mismatch; arxiv_id, observed 2026-06-29T08:43:15.235293Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c9b9bdf9-7ed9-495c-b542-75333cb6dc10 · inbound

Incremental Learning in Mirror Flows cites this paper.

Incremental Learning in Mirror Flows Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 20

Resolution: metadata mismatch; arxiv_id, observed 2026-07-04T11:49:50.323221Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 5722c635-da17-4046-a543-86ce7abeff20 · inbound

Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics cites this paper.

Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 14

Resolution: verified exact; arxiv_id, observed 2026-06-30T07:24:21.767685Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 79880936-ea8e-46f1-8465-aee9ee8437c2 · inbound

Effective dynamics of the Sinkhorn algorithm in the regime of low entropy regularization cites this paper.

Effective dynamics of the Sinkhorn algorithm in the regime of low entropy regularization Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 132

Resolution: verified exact; arxiv_id, observed 2026-07-02T08:06:47.629924Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 30a82447-6799-47ae-b1a5-dddadb43daf8 · inbound

Optimal Learning Rate Scaling Depends on Data in Deep Scalar Linear Networks cites this paper.

Optimal Learning Rate Scaling Depends on Data in Deep Scalar Linear Networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 33

Resolution: verified exact; local_arxiv, observed 2026-07-10T16:07:20.385724Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 784de15a-c03b-4197-8571-f18e7530f66d · inbound

Singular perturbations and hierarchical learning in two-layer neural networks cites this paper.

Singular perturbations and hierarchical learning in two-layer neural networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

Reference 55

Resolution: unresolved; no resolver link, observed 2026-07-14T08:40:30.522134Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.