Pith Number

pith:DRDBSOJU

pith:2026:DRDBSOJUZP7HJ5RCM54YV5E62W

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The Geometric Structure of Models Learning Sparse Data

Ahmed Imtiaz Humayun, Randall Balestriero, Richard Baraniuk, Thomas Walker, T. Mitchell Roddenberry

Models succeed on sparse data by making their input-output Jacobians rank-one and perfectly aligned with each training point.

arxiv:2605.08464 v2 · 2026-05-08 · cs.LG

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Claims

C1strongest claim

normal-aligned classifiers -- whose input-output Jacobians are rank-one and align perfectly with the training data -- minimize the training objective under norm constraints and achieve maximal local robustness under a non-zero Jacobian constraint

C2weakest assumption

The assumption that success in the sparse regime is explained by normal alignment rather than other mechanisms, and that this alignment arises specifically from the feature-learning regime in continuous piecewise-affine networks (as described in the abstract when discussing power-diagram partitions).

C3one line summary

Normal alignment is the rank-one Jacobian structure that lets classifiers minimize loss and maximize local robustness in sparse regimes; the paper proves its optimality and uses it to create GrokAlign and RFAMs.

References

46 extracted · 46 resolved · 2 Pith anchors

[1] Tenenbaum, Vin de Silva, and John C 2000

[2] Roweis and Lawrence K 2000

[3] Dauphin, and David Lopez-Paz 2018

[4] CutMix: Regularization Strategy to Train Strong Classifiers with Localizable Features 2019

[5] Data Augmentation Using Random Image Cropping and Patching for Deep CNNs.IEEE Trans 2020

Formal links

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Receipt and verification

First computed	2026-05-20T00:01:43.020288Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1c46193934cbfe74f62267798af49ed5a521499f2dafb7129c80ef81d8bff2d0

Aliases

arxiv: 2605.08464 · arxiv_version: 2605.08464v2 · doi: 10.48550/arxiv.2605.08464 · pith_short_12: DRDBSOJUZP7H · pith_short_16: DRDBSOJUZP7HJ5RC · pith_short_8: DRDBSOJU

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/DRDBSOJUZP7HJ5RCM54YV5E62W \
  | 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: 1c46193934cbfe74f62267798af49ed5a521499f2dafb7129c80ef81d8bff2d0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "05a5e5d4b0216f5dbb4313e92ce1a8b57ea3950b29b5bf62dd042a4e57e6ae97",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-08T20:30:22Z",
    "title_canon_sha256": "77607c69e021cb880fe63eec12b930157c1c8fc22ab1213560a67c3184249fa3"
  },
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  "source": {
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    "kind": "arxiv",
    "version": 2
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