Pith Number

pith:PCRIFIFQ

pith:2026:PCRIFIFQ3AKX2NASO54YF2G3MG

not attested not anchored not stored refs pending

Exponential Approximation Rates and Parameter Efficiency of Learnable Bernstein Activations

Ibrahim Albool, Malak Gamal El-Din, Salma Elmalaki, Yasser Shoukry

Learnable Bernstein polynomial activations achieve approximation error decaying as O(n^{-L}) with network depth and polynomial order, exponentially faster than ReLU networks while remaining fully differentiable.

arxiv:2602.04264 v2 · 2026-02-04 · cs.LG · cs.AI · cs.NA · math.NA

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\usepackage{pith}
\pithnumber{PCRIFIFQ3AKX2NASO54YF2G3MG}

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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

their approximation error decays with the network depth L and the polynomial order n with a rate of O(n^{-L}), exponentially faster than the polynomial rate of ReLU architectures while remaining fully differentiable.

C2weakest assumption

The theoretical analysis assumes that the learnable Bernstein activations can be integrated into standard deep network architectures without introducing optimization difficulties that would negate the approximation benefits.

C3one line summary

Learnable Bernstein activations achieve exponential approximation rates O(n^{-L}) and deliver up to 99.9% parameter reduction while matching or exceeding ReLU performance on large physics datasets.

Receipt and verification

First computed	2026-05-18T03:09:23.943437Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

78a282a0b0d8157d3412777982e8db619aa4a10971f47739c25c4edf2c67c76a

Aliases

arxiv: 2602.04264 · arxiv_version: 2602.04264v2 · doi: 10.48550/arxiv.2602.04264 · pith_short_12: PCRIFIFQ3AKX · pith_short_16: PCRIFIFQ3AKX2NAS · pith_short_8: PCRIFIFQ

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/PCRIFIFQ3AKX2NASO54YF2G3MG \
  | 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: 78a282a0b0d8157d3412777982e8db619aa4a10971f47739c25c4edf2c67c76a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "daf496ce8d5fdc03fd988f3cbf5e60d5a7035e082ddce5807565ae7394c802f1",
    "cross_cats_sorted": [
      "cs.AI",
      "cs.NA",
      "math.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-02-04T06:55:55Z",
    "title_canon_sha256": "9dca2bdd3ff3db2aeb9324011d1269d146505ca4273ea572b9e32bb75e3657ef"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2602.04264",
    "kind": "arxiv",
    "version": 2
  }
}