Paper Citation Record · LEDGER

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model

As of 9 August 2026, this Paper Citation Record lists 39 of 39 outbound references and 3 inbound Pith citation observations for arXiv:2506.02664.

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

pith.paper-citation-record.v1
2506.02664 v1

Coverage vector

measured 39 of 39 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T11:28:30.317016Z

measured 42 of 42 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-09T06:31:02.800959+00:00

measured 3 of 3 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-06-27T14:08:33.542700Z

measured 1 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-08-05T02:28:24.338817Z

Reference resolution

39 of 39 outbound references displayed

verified exact0
verified fuzzy29
unresolved10
parse uncertain0
malformed identifier0
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External citation measurements

0
arxiv_reference, observed 2026-08-05T02:28:24.338817Z

Outbound references

Observation 12b171b8-2344-450f-965e-3c1371144474 · outbound

This paper cites Deep learning.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Deep learning

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:26.247557Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4bed6373-df5f-469e-a365-992f808701ca · outbound

This paper cites Multimodal deep learning.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Multimodal deep learning

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:36.098008Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation d0cb3d6a-9f4b-4b02-a984-c51afcf044ac · outbound

This paper cites Multimodal machine learning: A survey and taxonomy.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Multimodal machine learning: A survey and taxonomy

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:26.438704Z

Source-reported events for the cited work

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Observation e2aa8248-9440-4681-9a96-3fb7869687c5 · outbound

This paper cites Curriculum learning.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Curriculum learning

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:26.535972Z

Source-reported events for the cited work

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Observation 04ff8b99-555c-4274-b453-96d210a53feb · outbound

This paper cites Ben Arous, J.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Ben Arous, J

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:35.874826Z

Source-reported events for the cited work

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Observation 5b8d572b-abdb-425d-9f52-fcf1f69f08a3 · outbound

This paper cites Optimal shrinkage of eigenvalues in the spiked covariance model.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Optimal shrinkage of eigenvalues in the spiked covariance model

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:35.731128Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 13444d8d-6f7e-4840-be42-82f7878bc216 · outbound

This paper cites A statistical model for tensor PCA.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model A statistical model for tensor PCA

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:35.541681Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 18f22d6c-76e8-4962-ab2d-a178388798ca · outbound

This paper cites Statistical and compu- tational phase transitions in spiked tensor estimation.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Statistical and compu- tational phase transitions in spiked tensor estimation

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:35.386805Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 2ffcf406-9043-49e6-956b-9081b863e482 · outbound

This paper cites Tensor principal component analysis via sum-of-square proofs.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Tensor principal component analysis via sum-of-square proofs

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:35.230021Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 70238a8f-2776-4711-9348-0e983c788928 · outbound

This paper cites Wein, and Afonso S.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Wein, and Afonso S

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:35.074734Z

Source-reported events for the cited work

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Observation 28036061-8f5d-4ac8-a0f5-0f6ba03c0824 · outbound

This paper cites The Kikuchi hierarchy and tensor PCA.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model The Kikuchi hierarchy and tensor PCA

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:34.870534Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation eb5edb36-fdcd-4a43-a2d3-fd5b74134dd2 · outbound

This paper cites Algorithmic thresholds for tensor PCA.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Algorithmic thresholds for tensor PCA

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:34.676916Z

Source-reported events for the cited work

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Observation a3f40b04-5d71-4994-9dd3-451677b5480d · outbound

This paper cites The layered structure of tensor estimation and its mutual information.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model The layered structure of tensor estimation and its mutual information

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:34.538265Z

Source-reported events for the cited work

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Observation 222a42b2-63f8-48ba-945b-59444afab78c · outbound

This paper cites Statistical mechanics of low-rank tensor decomposition.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Statistical mechanics of low-rank tensor decomposition

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:34.333060Z

Source-reported events for the cited work

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Observation 88b7a069-fad4-4ceb-b14f-9bca21f47382 · outbound

This paper cites The staircase property: How hierarchical structure can guide deep learning.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model The staircase property: How hierarchical structure can guide deep learning

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:34.112442Z

Source-reported events for the cited work

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Observation 16a79474-3e26-4e98-a0e1-76d192ffc861 · outbound

This paper cites SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:33.937755Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 8e674ef7-712b-4281-aec5-0733e1f9ff65 · outbound

This paper cites Fundamental limits of weak learnability in high-dimensional multi-index models.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Fundamental limits of weak learnability in high-dimensional multi-index models

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:33.736245Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation ab4b724a-7d2e-4cc7-862c-39f8c11184c3 · outbound

This paper cites The high-dimensional asymptotics of first order methods with random data.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model The high-dimensional asymptotics of first order methods with random data

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:28.086086Z

Source-reported events for the cited work

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Observation f4c85d3f-ef5c-42fd-ad31-643f81366002 · outbound

This paper cites Information-theoretically optimal sparse PCA.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Information-theoretically optimal sparse PCA

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:33.590998Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 5d664236-4d75-442c-80ba-03db21937b5d · outbound

This paper cites Phase transition of the largest eigenvalue for non-null complex sample covariance matrices.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Phase transition of the largest eigenvalue for non-null complex sample covariance matrices

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:33.415729Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 2c2d53d1-b0a6-4d48-bec8-2de0994b321c · outbound

This paper cites Constrained low-rank matrix estimation: phase transitions, approximate message passing and applications.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Constrained low-rank matrix estimation: phase transitions, approximate message passing and applications

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:33.223937Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation d02d8ae1-c42f-4713-b22d-b9697083cbc3 · outbound

This paper cites Spherical 2+p Spin-Glass model: An exactly solvable model for Glass to Spin-Glass transition.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Spherical 2+p Spin-Glass model: An exactly solvable model for Glass to Spin-Glass transition

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:33.003613Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 1b49f750-87a5-482d-939e-920c398d5976 · outbound

This paper cites Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.866524Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation ba9718b1-4e6f-4591-ac28-98f39bcaf08c · outbound

This paper cites Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.703661Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 89c08091-de24-4e90-8eb0-9adb7bc3c4a2 · outbound

This paper cites Passed & Spurious: Descent Algorithms and Local Minima in Spiked Matrix-Tensor Models.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Passed & Spurious: Descent Algorithms and Local Minima in Spiked Matrix-Tensor Models

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.600751Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 82790fff-b239-4cb3-8644-54dffe466d50 · outbound

This paper cites Who is Afraid of Big Bad Minima? Analysis of gradient-flow in spiked matrix-tensor models.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Who is Afraid of Big Bad Minima? Analysis of gradient-flow in spiked matrix-tensor models

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.438574Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 980b768d-fa56-4d0b-bff4-8b2342a45661 · outbound

This paper cites Donoho, Arian Maleki, and Andrea Montanari.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Donoho, Arian Maleki, and Andrea Montanari

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.324757Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 2388fabc-cf6a-457c-9636-6be54a4994e1 · outbound

This paper cites State evolution for general approximate message passing algorithms, with applications to spatial coupling.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model State evolution for general approximate message passing algorithms, with applications to spatial coupling

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.138858Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation a8f0e19e-63d1-42d6-bf21-ef69d6576327 · outbound

This paper cites Sampling from mean-field Gibbs measures via diffusion processes.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Sampling from mean-field Gibbs measures via diffusion processes

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:29.347327Z

Source-reported events for the cited work

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Observation 0ee974ed-8e4b-4410-9a63-5f99f3af8ce6 · outbound

This paper cites The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model The merged-staircase property: a necessary and nearly sufficient condition for sgd learning of sparse functions on two-layer neural networks

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:32.014497Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 050ff04e-cd0a-4c7a-8b2d-8d6f606c4ad1 · outbound

This paper cites A survey on deep multimodal learning for computer vision: advances, trends, applications, and datasets.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model A survey on deep multimodal learning for computer vision: advances, trends, applications, and datasets

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:31.822108Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 26b553ef-5d9a-48b8-9263-93d85d80a018 · outbound

This paper cites Simultaneous Dimensionality Reduction: A Data Efficient Approach for Multimodal Representations Learning.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Simultaneous Dimensionality Reduction: A Data Efficient Approach for Multimodal Representations Learning

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:29.691447Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 15149a54-de70-4d69-bdc0-56807e852ea6 · outbound

This paper cites Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions

Reference 33

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:29.786427Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 87606277-6808-4c6e-8a0a-07af517097b3 · outbound

This paper cites Multimodal data integration and cross-modal querying via orchestrated approximate message passing.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Multimodal data integration and cross-modal querying via orchestrated approximate message passing

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-07T11:28:29.877756Z

Source-reported events for the cited work

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Observation 105d6eca-7b24-443a-a136-cd715c328973 · outbound

This paper cites A unifying tutorial on approximate message passing.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model A unifying tutorial on approximate message passing

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:31.595686Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation f19178dd-ee9b-4f91-9a13-f0a7d4824616 · outbound

This paper cites Disordered systems insights on computational hardness.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Disordered systems insights on computational hardness

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:31.422711Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 779fd875-fbef-46e5-8829-b3f7925c1f2c · outbound

This paper cites ∼ N(0, 1), and we recall [g(qt)]1 = α2 ∆m qt 2 + α3α4 ∆t qt 3qt.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model ∼ N(0, 1), and we recall [g(qt)]1 = α2 ∆m qt 2 + α3α4 ∆t qt 3qt

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:28:31.221914Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 34cf286c-005d-4f62-90a3-0f9133627eba · outbound

This paper cites an unresolved cited work.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Unresolved cited work

Reference 38

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:28:31.011121Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation a62d0de1-f3f9-40b3-bcba-3dd6098761e5 · outbound

This paper cites an unresolved cited work.

Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model Unresolved cited work

Reference 39

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:28:30.844007Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Pith citing papers

Observation 69aa46d0-23fa-43ec-b8a7-5116ac76853b · inbound

Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning cites this paper.

Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model

Reference 71

Resolution: verified exact; arxiv_id, observed 2026-05-14T19:42:53.997267Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 32f08667-98bb-48cc-a92b-a5bc0ebc09be · inbound

Optimal Spectral Algorithms for Correlated Two-view Models in High Dimensions cites this paper.

Optimal Spectral Algorithms for Correlated Two-view Models in High Dimensions Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model

Reference 37

Resolution: verified exact; arxiv_id, observed 2026-05-20T02:37:59.303084Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.

Observation 939304f5-a852-4fae-94d5-0bf163fedebf · inbound

When to Align, When to Predict: A Phase Diagram for Multimodal Learning cites this paper.

When to Align, When to Predict: A Phase Diagram for Multimodal Learning Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model

Reference 35

Resolution: metadata mismatch; arxiv_id, observed 2026-06-27T14:10:57.767622Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-09T06:31:02.800959+00:00.