Paper Citation Record · LEDGER

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks

As of 11 August 2026, this Paper Citation Record lists 26 of 26 outbound references and 1 inbound Pith citation observation for arXiv:2501.09137.

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

pith.paper-citation-record.v1
2501.09137 v2

Coverage vector

measured 26 of 26 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-10T20:26:43.530125Z

measured 27 of 27 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-29T13:20:54.303605Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-06-29T13:23:28.017867Z

Reference resolution

26 of 26 outbound references displayed

verified exact0
verified fuzzy15
unresolved11
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation a3c0857e-9576-4883-b16a-138151270d33 · outbound

This paper cites write newline.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks write newline

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.412162Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c5f48a3d-e7f8-4ddc-a62f-e1616fe57885 · outbound

This paper cites T., Suarez, F., and Zhang, Y.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks T., Suarez, F., and Zhang, Y

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.941693Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation c5b2b8e0-c4fb-4cb4-b295-8567443c7197 · outbound

This paper cites A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.421051Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation da037326-655f-489c-9d80-f24bc6feaf81 · outbound

This paper cites P., Selman, B., and Weinberger, K.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks P., Selman, B., and Weinberger, K

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.928142Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation e59cfc15-b3c9-487d-88e6-bf090f02520f · outbound

This paper cites Optimization Methods for Large-Scale Machine Learning.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Optimization Methods for Large-Scale Machine Learning

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.430882Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f5d9d63a-5642-4d59-ba11-4f91cf4fb02d · outbound

This paper cites and Bruna, J.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks and Bruna, J

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.908686Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 5ec7451e-75d1-4dd3-9dad-51763c0f20e3 · outbound

This paper cites Z., and Talwalkar, A.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Z., and Talwalkar, A

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.889619Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 98fce254-a364-4024-8a8d-cf374f5f1177 · outbound

This paper cites Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.873290Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation a66da8c3-f9f6-4ce4-8668-f04cfd0b3d25 · outbound

This paper cites and Schmidhuber, J.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks and Schmidhuber, J

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.857854Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 836d0d80-4c5a-441e-ba53-19e71e36882a · outbound

This paper cites The Break-Even Point on Optimization Trajectories of Deep Neural Networks.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks The Break-Even Point on Optimization Trajectories of Deep Neural Networks

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.454094Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fb4d54e9-6740-4744-9a80-a8b7f500874f · outbound

This paper cites S., Mudigere, D., Nocedal, J., Smelyanskiy, M., and Tang, P.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks S., Mudigere, D., Nocedal, J., Smelyanskiy, M., and Tang, P

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.836855Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation be8bfd10-f67b-4c79-a432-26eb978412e4 · outbound

This paper cites B., and Müller, K.-R.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks B., and Müller, K.-R

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.817831Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 7f64b7da-712d-495f-8161-69a21ad71df1 · outbound

This paper cites The large learning rate phase of deep learning: the catapult mechanism.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks The large learning rate phase of deep learning: the catapult mechanism

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.466024Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9d8049ef-d133-44ef-920f-a71bdfe7ac22 · outbound

This paper cites Towards explaining the regularization effect of initial large learning rate in training neural networks.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Towards explaining the regularization effect of initial large learning rate in training neural networks

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.802916Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 4181a5dd-1f4d-4c31-a8c4-174f953c33b4 · outbound

This paper cites M., Rauhut, H., and Terstiege, U.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks M., Rauhut, H., and Terstiege, U

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.784615Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation b2a8ab38-6d47-4557-8a67-bc6e331773df · outbound

This paper cites The Effect of Network Width on Stochastic Gradient Descent and Generalization : an Empirical Study.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks The Effect of Network Width on Stochastic Gradient Descent and Generalization : an Empirical Study

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.767367Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation ca90ba2d-03ac-4964-967f-6c2b1c79ed06 · outbound

This paper cites an unresolved cited work.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-10T20:26:43.752819Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 2dd077db-079a-4f7d-bebe-0efc2e47a981 · outbound

This paper cites Exact solutions to the nonlinear dynamics of learning in deep linear neural networks.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Exact solutions to the nonlinear dynamics of learning in deep linear neural networks

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.492055Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fef5958b-0a7e-4721-b896-35d878e34995 · outbound

This paper cites A Bayesian Perspective on Generalization and Stochastic Gradient Descent.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks A Bayesian Perspective on Generalization and Stochastic Gradient Descent

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.497375Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a9308a71-ecfb-4aa5-bc96-0aba767f8dbf · outbound

This paper cites D., and Vidal, R.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks D., and Vidal, R

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.737100Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation a26044fc-ce5c-4f30-bf6e-eeab91f6b694 · outbound

This paper cites Large Learning Rate Tames Homogeneity : Convergence and Balancing Effect.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Large Learning Rate Tames Homogeneity : Convergence and Balancing Effect

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.717299Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation edd11cbf-48dc-408a-9fe7-0f43af6fa797 · outbound

This paper cites Three Mechanisms of Feature Learning in a Linear Network.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Three Mechanisms of Feature Learning in a Linear Network

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.512820Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0952d60f-06b8-453c-b4c8-c883720e3fd0 · outbound

This paper cites Linear convergence of gradient descent for finite width over-parametrized linear networks with general initialization.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Linear convergence of gradient descent for finite width over-parametrized linear networks with general initialization

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.700426Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 7aa214f3-1596-4b92-a4ec-3480c1750796 · outbound

This paper cites @esa (Ref.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks @esa (Ref

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.521281Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5e389e42-4f56-4eaa-bd43-7f8f4921c8b6 · outbound

This paper cites an unresolved cited work.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks Unresolved cited work

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-10T20:26:43.525592Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b494e2f5-df1c-419d-8b51-6f59a2a6230a · outbound

This paper cites ,# (7),01444 '9=82<.342C 2! !22222222222222222222222222222222222222222222222222.

Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks ,# (7),01444 '9=82<.342C 2! !22222222222222222222222222222222222222222222222222

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T20:26:43.666612Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Pith citing papers

Observation f1271a0b-d08e-4163-8536-08678091a4bd · inbound

Do Deep Networks Forget Initialization? A Forgetting-Time View of Practical Inductive Bias cites this paper.

Do Deep Networks Forget Initialization? A Forgetting-Time View of Practical Inductive Bias Gradient Descent Converges Linearly to Flatter Minima than Gradient Flow in Shallow Linear Networks

Reference 42

Resolution: verified exact; arxiv_id, observed 2026-06-29T13:23:28.020113Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.