Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks | Paper Citation Record

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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This paper cites Mehta, M.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Mehta, M

Reference 2

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This paper cites Bahri, J.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Bahri, J

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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This paper cites Seroussi, G.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Seroussi, G

Reference 5

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

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This paper cites Menon, The geometry of the deep linear network, inXIV Symposium on Probability and Stochastic Pro- cesses, edited by C.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Menon, The geometry of the deep linear network, inXIV Symposium on Probability and Stochastic Pro- cesses, edited by C

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This paper cites Zhang, A.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Zhang, A

Reference 9

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Kunin, G

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Zhang, Z

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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This paper cites Parameter Symmetry Potentially Unifies Deep Learning Theory.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Parameter Symmetry Potentially Unifies Deep Learning Theory

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This paper cites Goldenfeld,Lectures On Phase Transitions And The Renormalization Group(CRC Press, 1992).

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Goldenfeld,Lectures On Phase Transitions And The Renormalization Group(CRC Press, 1992)

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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This paper cites Ziyin, B.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Ziyin, B

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Ziyin and M

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This paper cites Ziyin, Symmetry induces structure and constraint of learning, inForty-first International Conference on Ma- chine Learning(2024).

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Ziyin, Symmetry induces structure and constraint of learning, inForty-first International Conference on Ma- chine Learning(2024)

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Applications of Statistical Field Theory in Deep Learning

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks There Will Be a Scientific Theory of Deep Learning

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Baldi and K

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Fukumizu, Dynamics of Batch Learning in Multilayer Neural Networks, inICANN 98, edited by L

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This paper cites Kawaguchi, Deep Learning without Poor Local Min- ima, inAdvances in Neural Information Processing Systems, Vol.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Kawaguchi, Deep Learning without Poor Local Min- ima, inAdvances in Neural Information Processing Systems, Vol

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Wendin and C

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Lindsey and G

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks An entropy formula for the Deep Linear Network

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Scarvelis and J

Reference 38

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Understanding Learning Invariance in Deep Linear Networks

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Wang and A

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Chechik, A

Reference 41

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Arora, N

Reference 42

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Trager, K

Reference 45

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Liang, L

Reference 47

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Altland and B

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Braun, C

Reference 50

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Roy and M

Reference 54

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Observation 3927575b-c943-4d74-93f8-c20b0d45c5f1 · outbound

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Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Yunis, K

Reference 55

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Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 97d5d4b8-a2d0-457b-88b2-54dc8a1a6ae4 · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 56

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.514642Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation f690d33b-633e-441b-b260-6707b71c8961 · outbound

This paper cites Li and H.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Li and H

Reference 57

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.498027Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation bcd2696d-1cdd-47ab-b624-f2d11b025b1c · outbound

This paper cites Bachtis, G.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Bachtis, G

Reference 58

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.480373Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 4ae5cb14-3c04-4f06-90d1-2cbfd84ed1c5 · outbound

This paper cites Low-rank bias, weight decay, and model merging in neural networks.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Low-rank bias, weight decay, and model merging in neural networks

Reference 59

Resolution: unresolved; no resolver link, observed 2026-08-15T14:39:45.753110Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0b5df035-4abb-4fb0-a391-7098f0e6c3fe · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 60

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.463749Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation c731aa5b-b139-459d-ab86-00754f8a2f4c · outbound

This paper cites Sasagawa, On the finite escape phenomena for ma- trix Riccati equations, IEEE Transactions on Automatic Control27, 977 (1982).

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Sasagawa, On the finite escape phenomena for ma- trix Riccati equations, IEEE Transactions on Automatic Control27, 977 (1982)

Reference 61

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.447304Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 11367d12-b222-47a2-abfa-1e4f8ff4efe4 · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 62

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.430795Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation fa4e983c-f924-453d-a068-2c76af81f81d · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 63

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.414362Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation bbdbfd84-c63b-412f-84d1-11971d9b8317 · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 64

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.397952Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation ebc92b39-e78b-48f2-8269-f0dd3557f853 · outbound

This paper cites Here,Dis a diagonal matrix with entriesκi.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Here,Dis a diagonal matrix with entriesκi

Reference 65

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.380218Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 9398d0a8-2d5e-4f17-86e8-eb81b77bbe5f · outbound

This paper cites For 1 hidden layer (n= 2):The first path corresponds to the direction dictated by the leading singular value ofΣyx, which isη 1 with (right) singular vector⃗ r1 =⃗ e1.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks For 1 hidden layer (n= 2):The first path corresponds to the direction dictated by the leading singular value ofΣyx, which isη 1 with (right) singular vector⃗ r1 =⃗ e1

Reference 66

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.363112Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation ff0f51f5-0f13-40ae-92f6-8c03c895f3a7 · outbound

This paper cites To analyze the Hessian, we are working with column vectorization, where vecc(W) := (W :,1,W :,2,...), such that the Hessian becomes a matrix.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks To analyze the Hessian, we are working with column vectorization, where vecc(W) := (W :,1,W :,2,...), such that the Hessian becomes a matrix

Reference 67

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.346162Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 38a72d20-42f0-4378-9230-403b76989fa6 · outbound

This paper cites The rankrof the criticalW (i)’s is crucial to understand the Hessian spectrum.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks The rankrof the criticalW (i)’s is crucial to understand the Hessian spectrum

Reference 68

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.327821Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation ba174b2c-fb75-4534-acb8-067994fff8d0 · outbound

This paper cites The main point is that the flat directions andβ-directions do not depend on the model details as they are associated with (broken) symmetry transformations.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks The main point is that the flat directions andβ-directions do not depend on the model details as they are associated with (broken) symmetry transformations

Reference 69

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.309596Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 7ee51643-098f-4468-be48-c36909b99437 · outbound

This paper cites However, we should get access to some of the eigenvectors and eigenvalues (namely those that do not break the0-balance condition).

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks However, we should get access to some of the eigenvectors and eigenvalues (namely those that do not break the0-balance condition)

Reference 70

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.292152Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation bed2f43a-46f5-4555-a3a0-d15332ef1605 · outbound

This paper cites The special case of 1 hidden layer (n= 2) instead can also be approached differently by making use of the Riccati equation.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks The special case of 1 hidden layer (n= 2) instead can also be approached differently by making use of the Riccati equation

Reference 71

Resolution: verified fuzzy; raw_fallback, observed 2026-08-15T14:39:46.273965Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation 9dbfecfc-d20e-4514-abe2-6c60b0c75d44 · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 72

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.257896Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

Observation f0c55f28-e7c4-41b3-97be-5e999821a51a · outbound

This paper cites an unresolved cited work.

Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks Unresolved cited work

Reference 73

Resolution: unresolved; raw_fallback, observed 2026-08-15T14:39:46.242039Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.

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Reference resolution

Outbound references

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