Paper Citation Record · LEDGER

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective

As of 4 August 2026, this Paper Citation Record lists 43 of 43 outbound references and 1 inbound Pith citation observation for arXiv:2601.06597.

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

pith.paper-citation-record.v1
2601.06597 v2

Coverage vector

measured 43 of 43 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-05-16T15:05:53.247993Z

measured 44 of 44 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-04T06:34:03.388597+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-01T15:35:24.756492Z

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

43 of 43 outbound references displayed

verified exact0
verified fuzzy41
unresolved2
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External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 4558508c-0486-4dc1-a2ea-4e6d18706f9f · outbound

This paper cites Reconciling modern machine- learning practice and the classical bias–variance trade-off.Proceedings of the National Academy of Sciences, 116(32):15849–15854, July 2019.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Reconciling modern machine- learning practice and the classical bias–variance trade-off.Proceedings of the National Academy of Sciences, 116(32):15849–15854, July 2019

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.598095Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation b8b698cf-7b91-4d42-b048-f0d18f5dbb77 · outbound

This paper cites In search of the real inductive bias: On the role of implicit regularization in deep learning.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective In search of the real inductive bias: On the role of implicit regularization in deep learning

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.595068Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 2b470e5e-b260-4e06-824e-cc18085c0dd7 · outbound

This paper cites The implicit bias of gradient descent on separable data.Journal of Machine Learning Research, 19(70):1–57.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective The implicit bias of gradient descent on separable data.Journal of Machine Learning Research, 19(70):1–57

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.606186Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation f5971906-6674-4a14-9826-8a1e1568cf61 · outbound

This paper cites On the implicit bias in deep-learning algorithms.Communications of the ACM, 66(6):86–93.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective On the implicit bias in deep-learning algorithms.Communications of the ACM, 66(6):86–93

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.592246Z

Source-reported events for the cited work

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Observation 31f0d703-195b-4729-93f4-de7683be4a8b · outbound

This paper cites The implicit bias of gradient descent on nonseparable data.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective The implicit bias of gradient descent on nonseparable data

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.603301Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 8f5d545c-3913-45da-92d2-13309c4fb49a · outbound

This paper cites Gradient descent maximizes the margin of homogeneous neural networks.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Gradient descent maximizes the margin of homogeneous neural networks

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.625224Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 316545b6-779a-42f0-a689-95de3460aea2 · outbound

This paper cites Schapire, and Matus Telgarsky.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Schapire, and Matus Telgarsky

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.622400Z

Source-reported events for the cited work

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Observation 4fbcfe1c-9afe-447a-ba3c-65ef67c2b1a7 · outbound

This paper cites Implicit bias of gradient descent for logistic regression at the edge of stability.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Implicit bias of gradient descent for logistic regression at the edge of stability

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.620101Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 78b0d7df-1f9d-43cd-8a67-4709a350cebb · outbound

This paper cites The implicit bias of gradient descent on separable multiclass data.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective The implicit bias of gradient descent on separable multiclass data

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.617272Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 3bb0058a-a060-49b0-9cad-838360c67c7c · outbound

This paper cites A unifying view on implicit bias in training linear neural networks.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective A unifying view on implicit bias in training linear neural networks

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.614504Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 99386dc1-7100-4111-bda2-a581f85a61db · outbound

This paper cites Characterizing implicit bias in terms of optimization geometry.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Characterizing implicit bias in terms of optimization geometry

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.600709Z

Source-reported events for the cited work

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Observation 02c21d1f-2db7-4805-bfe6-ff13c63a4618 · outbound

This paper cites Implicit regularization in deep matrix factorization.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Implicit regularization in deep matrix factorization

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.611803Z

Source-reported events for the cited work

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Observation 085625dc-2737-419c-8ec4-eb894e915c85 · outbound

This paper cites Implicit regularization of discrete gradient dynamics in linear neural networks.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Implicit regularization of discrete gradient dynamics in linear neural networks

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.608879Z

Source-reported events for the cited work

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Observation ff9218a9-43f9-47de-a398-ad86cde80575 · outbound

This paper cites Gradient descent for deep matrix factorization: Dynamics and implicit bias towards low rank.Applied and Computational Harmonic Analysis, 68:101595.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Gradient descent for deep matrix factorization: Dynamics and implicit bias towards low rank.Applied and Computational Harmonic Analysis, 68:101595

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.589672Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation b3019ee4-68be-451f-8836-4efb1adb18ed · outbound

This paper cites Dynamics in deep classifiers trained with the square loss: Normalization, low rank, neural collapse, and generalization bounds.Research, 6:0024.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Dynamics in deep classifiers trained with the square loss: Normalization, low rank, neural collapse, and generalization bounds.Research, 6:0024

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.587140Z

Source-reported events for the cited work

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Observation f5fc438e-39f0-476b-a5ba-70f8bead96f1 · outbound

This paper cites Implicit regularization in deep learning may not be explainable by norms.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Implicit regularization in deep learning may not be explainable by norms

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.584460Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation c343e0f9-10e3-4d04-b4b5-d40e3b7120a9 · outbound

This paper cites What happens after SGD reaches zero loss? – a mathematical framework.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective What happens after SGD reaches zero loss? – a mathematical framework

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.581811Z

Source-reported events for the cited work

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Observation db58c945-b475-4e8e-9f58-d622c1d7ed19 · outbound

This paper cites Implicit bias of deep linear networks in the large learning rate phase.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Implicit bias of deep linear networks in the large learning rate phase

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.579688Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation f789ccd1-faf8-445d-895f-197ac5fc4eb5 · outbound

This paper cites an unresolved cited work.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-05-16T15:08:02.577093Z

Source-reported events for the cited work

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Observation 53b36b0a-25d2-4fc1-8995-91116b1869bd · outbound

This paper cites PhD thesis, Toyota Technological Institute at Chicago.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective PhD thesis, Toyota Technological Institute at Chicago

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.574658Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 2d65080c-af37-4488-87c0-846d7e7d17d6 · outbound

This paper cites Implicit bias of gradient descent for wide two-layer neural networks trained with the logistic loss.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Implicit bias of gradient descent for wide two-layer neural networks trained with the logistic loss

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.572348Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation e4261c9f-64c7-4070-a111-c3e45a154e54 · outbound

This paper cites Stochastic gradient descent as approximate Bayesian inference.Journal of Machine Learning Research, 18(134):1–35.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Stochastic gradient descent as approximate Bayesian inference.Journal of Machine Learning Research, 18(134):1–35

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.569541Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 2214ca9e-5a86-4a33-9946-b44d62eab83e · outbound

This paper cites Stochastic modified equations and adaptive stochastic gradient algorithms.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Stochastic modified equations and adaptive stochastic gradient algorithms

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.567042Z

Source-reported events for the cited work

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Observation f6ce314b-b44a-4ae9-8f73-915192804c3a · outbound

This paper cites Stochastic modified equations and dynamics of stochastic gradient algorithms I: Mathematical foundations.Journal of Machine Learning Research, 20(40):1–47.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Stochastic modified equations and dynamics of stochastic gradient algorithms I: Mathematical foundations.Journal of Machine Learning Research, 20(40):1–47

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.564206Z

Source-reported events for the cited work

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Observation 77c06635-d19e-4bbc-9ece-5121913c3542 · outbound

This paper cites Theory of deep learning IIb: Optimization properties of SGD.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Theory of deep learning IIb: Optimization properties of SGD

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.561294Z

Source-reported events for the cited work

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Observation c6864c4b-193b-437f-9112-f7d770fa5653 · outbound

This paper cites A Bayesian perspective on generalization and stochastic gradient descent.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective A Bayesian perspective on generalization and stochastic gradient descent

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.558740Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 14410bd6-8a0d-4faa-beb4-4e204cc1f7bf · outbound

This paper cites A diffusion theory for deep learning dynamics: Stochastic gradient descent exponentially favors flat minima.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective A diffusion theory for deep learning dynamics: Stochastic gradient descent exponentially favors flat minima

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.555932Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 3eb89ed5-d1d5-4dd3-a05f-af4e269ae4ff · outbound

This paper cites Topological invariance and breakdown in learning.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Topological invariance and breakdown in learning

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.550138Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 77982a00-1354-4ce0-8aed-31e4e8302094 · outbound

This paper cites Neural thermodynamics: Entropic forces in deep and universal representation learning.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Neural thermodynamics: Entropic forces in deep and universal representation learning

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.552804Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 4654c433-abe0-45e4-b1cf-30347639d5cf · outbound

This paper cites Parameter symmetry and noise equilibrium of stochastic gradient descent.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Parameter symmetry and noise equilibrium of stochastic gradient descent

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.547522Z

Source-reported events for the cited work

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Observation 31fef6e4-7465-498f-bb06-710e9364dcad · outbound

This paper cites Symmetry induces structure and constraint of learning.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Symmetry induces structure and constraint of learning

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.509146Z

Source-reported events for the cited work

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Observation 2eeb9125-dc46-4885-924f-2a9d21d40ebf · outbound

This paper cites Parameter symmetry potentially unifies deep learning theory.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Parameter symmetry potentially unifies deep learning theory

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.544562Z

Source-reported events for the cited work

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Observation 580e12b0-955a-4608-a809-084ba7cf9fc1 · outbound

This paper cites Cambridge University Press, Cambridge, UK.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Cambridge University Press, Cambridge, UK

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.541777Z

Source-reported events for the cited work

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Observation 6987b6c4-bf7b-42e5-81f9-b6c7d9cfc096 · outbound

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Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Unresolved cited work

Reference 34

Resolution: unresolved; raw_fallback, observed 2026-05-16T15:08:02.538988Z

Source-reported events for the cited work

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Observation 4da4b0d8-74db-4d59-9c4c-88725ecb8986 · outbound

This paper cites Intrinsic statistics on Riemannian manifolds: Basic tools for geometric mea- surements.Journal of Mathematical Imaging and Vision, 25(1):127–154.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Intrinsic statistics on Riemannian manifolds: Basic tools for geometric mea- surements.Journal of Mathematical Imaging and Vision, 25(1):127–154

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.536861Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation fcb00998-cd8d-4d4f-8e2d-ce55776d1f4f · outbound

This paper cites Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions.Statistica Sinica, 20(1):1–100.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Intrinsic shape analysis: Geodesic principal component analysis for Riemannian manifolds modulo Lie group actions.Statistica Sinica, 20(1):1–100

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.534005Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 575f0f12-72b3-48c6-9280-facee61593db · outbound

This paper cites Classical statistical mechanics of constraints: A theorem and applications to polymers.The Journal of Chemical Physics, 69(4):1527–1537.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Classical statistical mechanics of constraints: A theorem and applications to polymers.The Journal of Chemical Physics, 69(4):1527–1537

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.531355Z

Source-reported events for the cited work

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Observation 82af91f5-e329-4acc-a3d1-ceb71baffd8c · outbound

This paper cites Imperial College Press, London.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Imperial College Press, London

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.528529Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 8438e8fd-77ff-4e09-92c7-d79c46d1f4fa · outbound

This paper cites Numerical-integration of Cartesian equations of motion of a system with constraints – molecular-dynamics of N-alkanes.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Numerical-integration of Cartesian equations of motion of a system with constraints – molecular-dynamics of N-alkanes

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.525460Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation a36f3954-7d41-4025-8770-b0b2d398447e · outbound

This paper cites Riemann manifold Langevin and Hamiltonian Monte Carlo methods.Journal of the Royal Statistical Society: Series B, 73(2):123–214.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Riemann manifold Langevin and Hamiltonian Monte Carlo methods.Journal of the Royal Statistical Society: Series B, 73(2):123–214

Reference 40

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.522695Z

Source-reported events for the cited work

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Observation ff0827ef-dbf9-4c51-bb05-6711de059906 · outbound

This paper cites Chrysos, YongtaoWu, RazvanPascanu, Philip Torr, andVolkan Cevher.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Chrysos, YongtaoWu, RazvanPascanu, Philip Torr, andVolkan Cevher

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.517997Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation 9f0f507d-92f4-47fe-b7a9-5a445ac978bc · outbound

This paper cites A survey on deep matrix factoriza- tions.Comput.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective A survey on deep matrix factoriza- tions.Comput

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.515166Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Observation f310fd19-6810-47db-959f-eda548f39e1b · outbound

This paper cites Springer, Berlin.

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective Springer, Berlin

Reference 43

Resolution: verified fuzzy; raw_fallback, observed 2026-05-16T15:08:02.512201Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.

Pith citing papers

Observation 5e63aa1f-f676-45b0-8a7e-0acb0f9824dc · inbound

PAC--Bayes Bounds on Quotient Parameter Spaces: Geometry-induced Implicit-Bias Priors cites this paper.

PAC--Bayes Bounds on Quotient Parameter Spaces: Geometry-induced Implicit-Bias Priors Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-01T15:35:24.756492Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.