Pith Number

pith:5BLQ2JLI

pith:2026:5BLQ2JLIEI5VQQA7JOK76IH3TA

not attested not anchored not stored refs pending

Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks

Bruno Loureiro, Jie Huang, Stefano Sarao Mannelli

Local minima in two-layer ReLU networks admit an exact low-dimensional representation via summary statistics.

arxiv:2604.09412 v2 · 2026-04-10 · stat.ML · cond-mat.dis-nn · cs.LG

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\usepackage{pith}
\pithnumber{5BLQ2JLIEI5VQQA7JOK76IH3TA}

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

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

We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space.

C2weakest assumption

The analysis assumes a realizable teacher-student setting with Gaussian covariates and the specific network form sum ReLU(w_k^T x), which may not hold for more general data distributions or deeper networks.

C3one line summary

Local minima of two-layer ReLU networks have an exact representation in terms of a few summary statistics that correspond to attractive fixed points of one-pass SGD, with overparameterization connecting them via flat directions.

Receipt and verification

First computed	2026-06-01T01:03:52.799059Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e8570d2568223b58401f4b95ff20fb9834af54541f32d74dd9429d3c49586448

Aliases

arxiv: 2604.09412 · arxiv_version: 2604.09412v2 · doi: 10.48550/arxiv.2604.09412 · pith_short_12: 5BLQ2JLIEI5V · pith_short_16: 5BLQ2JLIEI5VQQA7 · pith_short_8: 5BLQ2JLI

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/5BLQ2JLIEI5VQQA7JOK76IH3TA \
  | 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: e8570d2568223b58401f4b95ff20fb9834af54541f32d74dd9429d3c49586448

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2b457ee33e7d83e54e093d13c42311766f8d2c24a201b3cd3dc40cb9add9b499",
    "cross_cats_sorted": [
      "cond-mat.dis-nn",
      "cs.LG"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-04-10T15:26:00Z",
    "title_canon_sha256": "2758a56d67f00d43bd6fb4300fc3a3330478542021269a49b3f30e34303d05b3"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.09412",
    "kind": "arxiv",
    "version": 2
  }
}