Pith Number

pith:JC54ULOC

pith:2026:JC54ULOC2QYNFIOWG7JDK5BJLZ

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How does feature learning reshape the function space?

Bruno Loureiro, Fanghui Liu, Jo\~ao Lobo, Long Tran-Than

In high dimensions, one large gradient step on a two-layer network produces features whose distribution approximates a target-dependent spiked Gaussian covariance, inducing a data-adaptive kernel that reshapes the function space.

arxiv:2605.17718 v1 · 2026-05-18 · stat.ML · cs.LG

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Claims

C1strongest claim

We prove that, in the high-dimensional proportional regime, after a large gradient step the post-update feature distribution is well approximated by a target-dependent spiked Gaussian covariance. This induces a data-adaptive kernel that reshapes the function space and modifies its spectral structure.

C2weakest assumption

The analysis assumes the high-dimensional proportional regime (n, d → ∞ with n/d fixed) together with a sufficiently large gradient step size that allows the post-update feature distribution to be approximated by the spiked Gaussian form; if this regime or step-size condition fails to hold, the claimed approximation and resulting kernel reshaping do not necessarily follow.

C3one line summary

In the high-dimensional proportional regime, a large gradient step on a two-layer network induces a target-dependent spiked Gaussian covariance on the features, yielding a data-adaptive kernel that amplifies target-aligned eigenvalues and mixes leading eigenfunctions.

References

300 extracted · 300 resolved · 2 Pith anchors

[1] Charles R. Harris and K. Jarrod Millman and St. Array programming with. 2020 , month = sep, journal =. doi:10.1038/s41586-020-2649-2 , publisher = 2020 · doi:10.1038/s41586-020-2649-2

[2] and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and

[3] Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V. and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P. and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and

[4] PyTorch: An Imperative Style, High-Performance Deep Learning Library , url =

[5] Journal of Machine Learning Research , year =

Receipt and verification

First computed	2026-05-20T00:04:54.553700Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

48bbca2dc2d430d2a1d637d23574295e7e44ffa4f6e31843a2f7ae6cfb809efb

Aliases

arxiv: 2605.17718 · arxiv_version: 2605.17718v1 · doi: 10.48550/arxiv.2605.17718 · pith_short_12: JC54ULOC2QYN · pith_short_16: JC54ULOC2QYNFIOW · pith_short_8: JC54ULOC

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JC54ULOC2QYNFIOWG7JDK5BJLZ \
  | 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: 48bbca2dc2d430d2a1d637d23574295e7e44ffa4f6e31843a2f7ae6cfb809efb

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "04dfd05f40b0f909587a10481e6fcc0d13e1e881075ad72eeca51348ce04fadd",
    "cross_cats_sorted": [
      "cs.LG"
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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-05-18T00:41:20Z",
    "title_canon_sha256": "3d6dbf9d303aa6cc243e652f2e1946ff3b62a373824bc5c5037c7fb308bc4d4c"
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  "source": {
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    "kind": "arxiv",
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