Paper Citation Record · LEDGER

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics

As of 12 August 2026, this Paper Citation Record lists 38 of 38 outbound references and 2 inbound Pith citation observations for arXiv:2412.05144.

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

pith.paper-citation-record.v1
2412.05144 v3

Coverage vector

measured 38 of 38 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-11T20:58:06.329477Z

measured 40 of 40 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00

measured 2 of 2 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-06T17:06:11.036615Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-07-03T09:07:47.415051Z

Reference resolution

38 of 38 outbound references displayed

verified exact0
verified fuzzy32
unresolved6
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External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 199d6f55-f3cb-4969-97dc-528fe8161a19 · outbound

This paper cites On the Inductive Bias of Neural Tangent Kernels.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics On the Inductive Bias of Neural Tangent Kernels

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.233066Z

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Observation d385873b-b24f-4709-a7fa-1a3e54dc30ce · outbound

This paper cites The Random Feature Method for Time-Dependent Problems.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics The Random Feature Method for Time-Dependent Problems

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.214296Z

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Observation 4e57e284-fe89-4f32-b25a-b9995e50a2f4 · outbound

This paper cites Optimization of Random Feature Method in the High- Precision Regime.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Optimization of Random Feature Method in the High- Precision Regime

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.194972Z

Source-reported events for the cited work

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Observation 6c726a6b-0e47-4a57-a7b7-aa884cfaf534 · outbound

This paper cites A Generalized Neural Tangent Kernel Analysis for Two-layer Neural Networks.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics A Generalized Neural Tangent Kernel Analysis for Two-layer Neural Networks

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.177345Z

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Observation 00bbd540-178b-4a37-8661-f54711e55eb3 · outbound

This paper cites Sharp Minima Can Gener- alize For Deep Nets.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Sharp Minima Can Gener- alize For Deep Nets

Reference 5

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Source-reported events for the cited work

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Observation 6eb3d5ee-af5b-4079-a1d2-93689f731b11 · outbound

This paper cites Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.138222Z

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Observation 92a21d70-2849-45d3-83dc-fa589048503a · outbound

This paper cites Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.107366Z

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Observation 83604192-e253-4064-98fe-74d4365ee7c0 · outbound

This paper cites The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.082531Z

Source-reported events for the cited work

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Observation 5867b15f-d029-4e16-a4ef-317443943ff7 · outbound

This paper cites Understanding the difficulty of training deep feedforward neural networks.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Understanding the difficulty of training deep feedforward neural networks

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.057964Z

Source-reported events for the cited work

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Observation 383c7646-2d98-4e48-ba0e-5c62ef4f505e · outbound

This paper cites ReLU deep neural networks from the hierarchical basis perspective.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics ReLU deep neural networks from the hierarchical basis perspective

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:07.034468Z

Source-reported events for the cited work

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Observation b96eb785-d943-4f3c-932d-f0692e0e0a7c · outbound

This paper cites Relu Deep Neural Networks and Linear Finite Elements.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Relu Deep Neural Networks and Linear Finite Elements

Reference 11

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Source-reported events for the cited work

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Observation 11cb9f93-d1cd-4c20-a94f-b172feecfa9c · outbound

This paper cites Flat Minima.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Flat Minima

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.983017Z

Source-reported events for the cited work

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Observation c58464d6-18b2-4f2d-bcd2-7ccffeb84aa5 · outbound

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$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Unresolved cited work

Reference 13

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Source-reported events for the cited work

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Observation b1e53fbb-6dc0-409b-bb54-395879a8f108 · outbound

This paper cites Approximation capabilities of multilayer feedforward networks.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Approximation capabilities of multilayer feedforward networks

Reference 14

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Observation 38c83707-3fe0-4041-abc3-2dddbabc6935 · outbound

This paper cites Universal Approximation using Incre- mental Constructive Feedforward Networks with Random Hidden Nodes.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Universal Approximation using Incre- mental Constructive Feedforward Networks with Random Hidden Nodes

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.919227Z

Source-reported events for the cited work

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Observation bcf7791f-6e92-4eca-b97c-5a7d909f44f1 · outbound

This paper cites Extreme learning machine: Theory and applications.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Extreme learning machine: Theory and applications

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.892686Z

Source-reported events for the cited work

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Observation 99ffa99e-99ed-426e-ba3a-953944e9accf · outbound

This paper cites Neural Tangent Kernel: Convergence and Generalization in Neural Networks.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Neural Tangent Kernel: Convergence and Generalization in Neural Networks

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.868969Z

Source-reported events for the cited work

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Observation 225fe0c6-9255-4402-a00a-4cbda0f3cbf9 · outbound

This paper cites On large-batch training for deep learning: Generalization gap and sharp minima.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics On large-batch training for deep learning: Generalization gap and sharp minima

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.843446Z

Source-reported events for the cited work

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Observation 9fb0374e-292c-4c95-b3be-cea5c0b38f34 · outbound

This paper cites Wight and Jia Zhao.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Wight and Jia Zhao

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.823006Z

Source-reported events for the cited work

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Observation 213187a6-8e48-400c-81a3-adfd16bbd16c · outbound

This paper cites Visualizing the Loss Landscape of Neural Nets.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Visualizing the Loss Landscape of Neural Nets

Reference 20

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Source-reported events for the cited work

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Observation 195fce02-92e2-486b-91d0-2b56458cef8b · outbound

This paper cites Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains.Communications in Computational Physics, 28(5):1970–2001, 2020.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains.Communications in Computational Physics, 28(5):1970–2001, 2020

Reference 21

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Observation 95cc8e94-274f-4152-8664-e08bbd5c705b · outbound

This paper cites Meethal, Anoop Kodakkal, Mohamed Khalil, Aditya Ghantasala, Birgit Obst, Kai- Uwe Bletzinger, and Roland W¨ uchner.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Meethal, Anoop Kodakkal, Mohamed Khalil, Aditya Ghantasala, Birgit Obst, Kai- Uwe Bletzinger, and Roland W¨ uchner

Reference 22

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Source-reported events for the cited work

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Observation 78d239ab-d8ea-4e5c-b322-407a43a71bc5 · outbound

This paper cites Mitusch, Simon W.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Mitusch, Simon W

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.733969Z

Source-reported events for the cited work

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Observation 7dfd4d69-208c-4916-a05e-bca94d9041e2 · outbound

This paper cites Nelsen and Andrew M.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Nelsen and Andrew M

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.711820Z

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Observation 6ad92995-b8f7-4f18-92ce-b59b23ff109f · outbound

This paper cites On the Spectral Bias of Neural Networks.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics On the Spectral Bias of Neural Networks

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.690246Z

Source-reported events for the cited work

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Observation b636619c-4b8c-4111-bf5d-8a399fa30083 · outbound

This paper cites Random features for large-scale kernel machines.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Random features for large-scale kernel machines

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.675808Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 7c840368-bf30-4da1-b68d-f571dd31c4cb · outbound

This paper cites Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-11T20:58:06.282674Z

Source-reported events for the cited work

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Observation c7eb4718-d2e7-4a2d-98a5-b66ded88ffce · outbound

This paper cites Ramuhalli, L.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Ramuhalli, L

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.650377Z

Source-reported events for the cited work

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Observation c8b02705-b0f5-4634-93dc-07b9a5d4960b · outbound

This paper cites DGM: A deep learning algorithm for solving partial differential equations.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics DGM: A deep learning algorithm for solving partial differential equations

Reference 29

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Observation 57b216e5-b2ae-4e70-b2d8-259c14a14b24 · outbound

This paper cites When and why PINNs fail to train: A neural tangent kernel perspective.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics When and why PINNs fail to train: A neural tangent kernel perspective

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.615081Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation effaeedc-1de5-45bd-a280-b13bc182a435 · outbound

This paper cites Multi-scale deep neural network (MscaleDNN) methods for oscillatory stokes flows in complex domains.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Multi-scale deep neural network (MscaleDNN) methods for oscillatory stokes flows in complex domains

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.466539Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation ec335bce-61d2-4673-8531-882a88edb199 · outbound

This paper cites Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.450553Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 7304f97c-8df4-4480-8562-b326374815fa · outbound

This paper cites Overview Frequency Principle/Spectral Bias in Deep Learning.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Overview Frequency Principle/Spectral Bias in Deep Learning

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.434833Z

Source-reported events for the cited work

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Observation b79a3a05-9b40-4252-8cd6-f4082bf293db · outbound

This paper cites Training Behavior of Deep Neural Network in Frequency Domain.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Training Behavior of Deep Neural Network in Frequency Domain

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.420676Z

Source-reported events for the cited work

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Observation cf471e87-88a9-47f4-80a2-171a2ad25ea4 · outbound

This paper cites Weak adversarial networks for high-dimensional partial differential equations.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Weak adversarial networks for high-dimensional partial differential equations

Reference 35

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Source-reported events for the cited work

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Observation d84149ba-7622-413c-ac70-23d679f64ed6 · outbound

This paper cites Zeiler and Rob Fergus.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Zeiler and Rob Fergus

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.397909Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 4ec6dd56-0add-46b9-93fa-f5c585c1d713 · outbound

This paper cites Why shallow networks struggle with approximating and learning high frequency: A numerical study, 2023.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Why shallow networks struggle with approximating and learning high frequency: A numerical study, 2023

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.385205Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation f427ead9-76c4-4295-9d6f-52251319e731 · outbound

This paper cites Transferable Neural Networks for Partial Differential Equations.

$\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics Transferable Neural Networks for Partial Differential Equations

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T20:58:06.368934Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Pith citing papers

Observation 6f58c896-8228-488c-9471-2bd9b890fab1 · inbound

Structured First-Layer Initialization Pre-Training Techniques to Accelerate Training Process Based on $\varepsilon$-Rank cites this paper.

Structured First-Layer Initialization Pre-Training Techniques to Accelerate Training Process Based on $\varepsilon$-Rank $\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics

Reference 35

Resolution: unresolved; no resolver link, observed 2026-08-06T17:06:11.036615Z

Source-reported events for the cited work

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Observation 4df569e9-0fe5-41ff-a23b-ce71e5dd38ae · inbound

RankVR: Low-Rank Structure Perception and Value Recalibration for Robust Composed Image Retrieval cites this paper.

RankVR: Low-Rank Structure Perception and Value Recalibration for Robust Composed Image Retrieval $\epsilon$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics

Reference 81

Resolution: verified exact; arxiv_id, observed 2026-07-03T09:07:47.416401Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.