Paper Citation Record · LEDGER

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression

As of 4 August 2026, this Paper Citation Record lists 39 of 39 outbound references and 1 inbound Pith citation observation for arXiv:2511.15841.

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

pith.paper-citation-record.v1
2511.15841 v3

Coverage vector

measured 39 of 39 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-03T21:25:28.331269Z

measured 40 of 40 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 2026-06-28T02:40:39.324301Z

measured 0 of 1 external citation measurements

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

Source: pith, observed 2026-07-02T11:56:55.619902Z

Reference resolution

39 of 39 outbound references displayed

verified exact0
verified fuzzy0
unresolved37
parse uncertain0
malformed identifier2
metadata mismatch0

External citation measurements

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Outbound references

Observation 18dc1219-6f23-49cc-8605-d0b777ecbd71 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 1

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Observation 9988e6e5-8bf6-4839-8b84-93066e8b3404 · outbound

This paper cites approximator.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression approximator

Reference 2

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Observation 7eaae0a5-3a05-470b-ba8d-7643cd807ad9 · outbound

This paper cites Denote ˜n=n/D F.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Denote ˜n=n/D F

Reference 3

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Observation 721526b8-07b6-4c7a-80ee-f3c2fb044bca · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation c9f1a1f6-8e51-4a41-9a2c-164e6dff5b4b · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation 368c3cf2-6b0b-46f7-8301-841ee06603a1 · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 6

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Observation a314b5fb-b86d-46f9-a3ed-a926ca863d45 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 7

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Observation 4fab79ff-6a72-4468-b944-1ab86420dd72 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 8

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Observation b6d9b368-3b65-4754-892a-bc8a06f62eab · outbound

This paper cites Let τ ∗ 2 := inf{τ >0 :A(τ)≤τ /4}.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Let τ ∗ 2 := inf{τ >0 :A(τ)≤τ /4}

Reference 9

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Observation f35a0bd5-9cc3-4280-a3c7-a7c717635231 · outbound

This paper cites where the second inequality is due to Lemma S2.1, the third one is due to Lemma S2.3.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression where the second inequality is due to Lemma S2.1, the third one is due to Lemma S2.3

Reference 10

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Observation f9f42f8b-11d7-42c7-acc5-881eb9edc029 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 11

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Observation 27cba5a6-f479-4740-83ba-a1e8bd5ea7a4 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 12

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Observation 33198f61-7916-4b02-b9b7-f2c2f7b8e63b · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 13

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Observation 2a1d2e44-18b5-46b0-ad57-e686642a4857 · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation 79155839-c8d6-4bf0-bcbb-474df40e7b8c · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation 3ce68df0-994f-4719-9eac-cb73ce3d0323 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 16

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Observation 0807c21c-4b7e-4a01-835c-e1c4d6447499 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 17

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Observation 049a8e47-8733-4298-81bd-e8695624cffe · outbound

This paper cites As a result, the first two cases can be combined and written as ˜n− 1 2+γ + ˜n − 1 2−s 1−1/m +sγ whenm≥2 andγ <2.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression As a result, the first two cases can be combined and written as ˜n− 1 2+γ + ˜n − 1 2−s 1−1/m +sγ whenm≥2 andγ <2

Reference 19

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Observation b461a166-f0df-48b5-baa2-fe389907ffd0 · outbound

This paper cites So the last case can be combined with part of the third case (i.e., whenm∈(1,2) andγ≥2) and written as ˜n− 1 2γ + ˜n − 1 2−s 1−1/m +γ whenγ≥2.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression So the last case can be combined with part of the third case (i.e., whenm∈(1,2) andγ≥2) and written as ˜n− 1 2γ + ˜n − 1 2−s 1−1/m +γ whenγ≥2

Reference 20

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Observation 6eb7c828-2723-46b7-856e-ddc0f9b7c983 · outbound

This paper cites Hence, the rate can be written as ˜n− 1 2+γ + ˜n − 1 2−s 1−1/m +sγ whenm∈(1,2), γ <2.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Hence, the rate can be written as ˜n− 1 2+γ + ˜n − 1 2−s 1−1/m +sγ whenm∈(1,2), γ <2

Reference 21

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Observation 5200d4f8-d4ea-483a-a702-37c07fb2c5c8 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 22

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Observation 7e24c7d1-d5bb-485b-a9a3-f0ee6b53f506 · outbound

This paper cites 22 In both cases, the desired bound follows.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression 22 In both cases, the desired bound follows

Reference 23

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Observation 1cef4e7f-efac-4729-aa57-51bb5378118b · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 24

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Observation 841b91f5-6306-4fad-b52e-2a3398cc4ee3 · outbound

This paper cites Consequently, we denote σs := sup f∈F ∥πs+1f−π sf∥ L1+κ(P) ≤4˜ϵ′ s.(S27) 30.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Consequently, we denote σs := sup f∈F ∥πs+1f−π sf∥ L1+κ(P) ≤4˜ϵ′ s.(S27) 30

Reference 25

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Observation 042909f7-8ae2-4396-aab6-bdbf2b8b170c · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 26

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Observation b3e93c9a-83b7-4ea8-a7ad-12c320aa5d19 · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

Reference 27

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Observation 0483346d-a7a6-4836-bfba-4538ff57acbb · outbound

This paper cites Takew n(x) =x 2.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Takew n(x) =x 2

Reference 28

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Observation e6e039b2-2827-4bd0-bf22-e06dcbbdddd5 · outbound

This paper cites Then, taking a functionf, such thatf 0,τ −f=g−f 0,τ , and we have: E[ℓ′ τ (y−f 0,τ (x))(f0,τ (x)−f(x))]<− 1 2 ∥g−f 0,τ ∥2 L2(P) =− 1 2 ∥f−f 0,τ ∥2 L2(P) , which is impossible.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Then, taking a functionf, such thatf 0,τ −f=g−f 0,τ , and we have: E[ℓ′ τ (y−f 0,τ (x))(f0,τ (x)−f(x))]<− 1 2 ∥g−f 0,τ ∥2 L2(P) =− 1 2 ∥f−f 0,τ ∥2 L2(P) , which is impossible

Reference 29

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Observation 244fde70-1ac3-4a45-8fb0-eb3357ce314a · outbound

This paper cites As a result, E[ℓ′ τ (y−f 2(x))2|x] =E[ℓ′ τ (ξ+f 0(x)−f 2(x))2|x]≤E[(ξ+f 0(x)−f 2(x))2 ∧τ 2|x] ≤E[(ξ+f 0(x)−f 2(x))2|x]∧τ 2 = E[ξ2|x] + (f0(x)−f 2(x))2 ∧τ 2 ≤(v2 + 4M2)∧τ 2.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression As a result, E[ℓ′ τ (y−f 2(x))2|x] =E[ℓ′ τ (ξ+f 0(x)−f 2(x))2|x]≤E[(ξ+f 0(x)−f 2(x))2 ∧τ 2|x] ≤E[(ξ+f 0(x)−f 2(x))2|x]∧τ 2 = E[ξ2|x] + (f0(x)−f 2(x))2 ∧τ 2 ≤(v2 + 4M2)∧τ 2

Reference 30

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Observation 6016472c-72d6-40ba-95b6-80803125f0ed · outbound

This paper cites Hence, we conclude that E h ℓτ (y−f 1(x))−ℓ τ (y−f 2(x)) 2i ≤2E h ((v2 + 4M2)∧τ 2)· |∆f(x)|2 i + 2M2∥∆f∥ 2 L2(P) =2 (v2 + 4M2)∧τ 2 +M 2 ∥∆f∥ 2 L2(P).

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Hence, we conclude that E h ℓτ (y−f 1(x))−ℓ τ (y−f 2(x)) 2i ≤2E h ((v2 + 4M2)∧τ 2)· |∆f(x)|2 i + 2M2∥∆f∥ 2 L2(P) =2 (v2 + 4M2)∧τ 2 +M 2 ∥∆f∥ 2 L2(P)

Reference 31

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Observation 4be83836-d7fc-4e60-9d32-898af6b35913 · outbound

This paper cites an unresolved cited work.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation 690498eb-06d6-4bf6-a19f-47a47653d59b · outbound

This paper cites Meanwhile, by Proposition S4.5, ∥ℓ∥L2(P) ≤ √ 2 τ (√v2 + 2M)∧τ+M ·c≤ √ 2 (√v2/τ+ 2M/τ)∧1 +M/τ ·c.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Meanwhile, by Proposition S4.5, ∥ℓ∥L2(P) ≤ √ 2 τ (√v2 + 2M)∧τ+M ·c≤ √ 2 (√v2/τ+ 2M/τ)∧1 +M/τ ·c

Reference 33

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Observation 6ca89a63-d1b1-4aa1-a7e6-34555df93724 · outbound

This paper cites 40 Denote ˜n=n/D Fn.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression 40 Denote ˜n=n/D Fn

Reference 34

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Observation e2cc6f55-9424-45f1-9796-29d799d0e87d · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation 8d9a98e5-c3cb-4f07-a7f2-0f4ae2bd7a3a · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation e57efdb1-c28b-443b-ac42-30606edc5357 · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation e8cdc5d8-015a-493d-8a2f-c2e809e051fe · outbound

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New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Unresolved cited work

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Observation 0aab741f-6c2a-4640-9a4f-93472ff795b6 · outbound

This paper cites Step 2.A (Envelope functionLF c).Let ∆f=f−f 0.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Step 2.A (Envelope functionLF c).Let ∆f=f−f 0

Reference 39

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Observation 6a440192-976b-4f53-afed-cfcd22716a9f · outbound

This paper cites Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression.

New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression

Reference 2008

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Pith citing papers

Observation 9db4192e-2deb-4d59-a106-ab60bfa28e19 · inbound

Mitigating the Curse of Dimensionality in Uniform Convergence of Deep Neural Networks via Smooth Activations cites this paper.

Mitigating the Curse of Dimensionality in Uniform Convergence of Deep Neural Networks via Smooth Activations New Empirical Process Tools and Their Applications to Robust Deep ReLU Networks and Phase Transitions for Nonparametric Regression

Reference 30

Resolution: metadata mismatch; local_arxiv, observed 2026-07-02T11:56:55.621214Z

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