Pith Number

pith:DMMM25JR

pith:2026:DMMM25JRWHB5BF6HFZCCG6P3H6

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A Fourier perspective on the learning dynamics of neural networks: from sample complexities to mechanistic insights

Claudia Merger, Fabiola Ricci, Sebastian Goldt

Online SGD cannot learn phase-only classification on isotropic high-dimensional inputs before order N cubed steps, but power-law spectra accelerate it substantially.

arxiv:2605.16913 v1 · 2026-05-16 · stat.ML · cond-mat.dis-nn · cond-mat.stat-mech · cs.LG · math.PR

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\usepackage{pith}
\pithnumber{DMMM25JRWHB5BF6HFZCCG6P3H6}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

For isotropic and high-dimensional inputs, classification based on phase information alone is a genuinely hard task: online SGD cannot distinguish the structured inputs from noise within n ≪ N³ steps, but needs at least n ≫ N³ log²N steps.

C2weakest assumption

The introduced synthetic data model for translation-invariant inputs faithfully captures the interaction between amplitudes, phases, and learning dynamics without introducing artifacts that invalidate the hardness result or the power-law acceleration claim.

C3one line summary

Neural networks prioritize amplitude over phase in Fourier space during training on translation-invariant data; power-law spectra accelerate phase learning despite not aiding classification.

References

69 extracted · 69 resolved · 0 Pith anchors

[1] et al.SGD on Neural Networks Learns Functions of Increasing ComplexityinAdvances in Neural Information Processing Systems32(2019), 3491–3501 2019

[2] Ingrosso, A. & Goldt, S. Data-driven emergence of convolutional structure in neural networks. Proceedings of the National Academy of Sciences119(2022) 2022

[3] & Goldt, S.Neural networks trained with SGD learn distributions of increasing complexityinInternational Conference on Machine Learning(2023), 28843–28863 2023

[4] & Goldt, S.A distributional simplicity bias in the learning dynamics of transformersinAdvances in Neural Information Processing Systems37(2024), 96207–96228 2024

[5] & Fern, X.Neural Networks Learn Statistics of Increasing Complexityin (arXiv, 2024) 2024

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:29.862957Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1b18cd7531b1c3d097c72e442379fb3facb2fa2df17c24134cd20b8db0c6f24c

Aliases

arxiv: 2605.16913 · arxiv_version: 2605.16913v1 · doi: 10.48550/arxiv.2605.16913 · pith_short_12: DMMM25JRWHB5 · pith_short_16: DMMM25JRWHB5BF6H · pith_short_8: DMMM25JR

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DMMM25JRWHB5BF6HFZCCG6P3H6 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 1b18cd7531b1c3d097c72e442379fb3facb2fa2df17c24134cd20b8db0c6f24c

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "192b42be698d2b273a2625dff2f2e745a58fa82ae6084e9fbe9494fa15d76360",
    "cross_cats_sorted": [
      "cond-mat.dis-nn",
      "cond-mat.stat-mech",
      "cs.LG",
      "math.PR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-05-16T10:03:41Z",
    "title_canon_sha256": "43eea5453ba470f353dc6b1cc1aae26ab974026a57d03cabe76876b04b100201"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16913",
    "kind": "arxiv",
    "version": 1
  }
}