Pith Number

pith:UDGTCHTA

pith:2026:UDGTCHTALB2ZZZ2JGXAIMDUKJ2

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Complex Stochastic Gradient Descent and Directional Bias in Reproducing Kernel Hilbert Spaces

Emeric Battaglia, Natanael Alpay

Complex SGD converges under the same assumptions as real SGD and extends directional bias to kernel regression in complex RKHS.

arxiv:2604.23017 v2 · 2026-04-24 · cs.LG · cs.NA · math.CV · math.NA

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\pithnumber{UDGTCHTALB2ZZZ2JGXAIMDUKJ2}

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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

We propose a variant of SGD (complex SGD) that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to GD as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for kernel regression problems.

C2weakest assumption

The assumptions that parallel those from the real setting (such as convexity, smoothness, or bounded variance) remain valid and sufficient when parameters and gradients are complex-valued, without additional analyticity requirements.

C3one line summary

Complex SGD converges without analyticity constraints and extends real-valued directional bias results to complex RKHS, with demonstrations on Fock and Hardy spaces.

Receipt and verification

First computed	2026-05-26T02:04:11.318890Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a0cd311e6058759ce74935c0860e8a4eb90604f63c9a83932c45f30f55563801

Aliases

arxiv: 2604.23017 · arxiv_version: 2604.23017v2 · doi: 10.48550/arxiv.2604.23017 · pith_short_12: UDGTCHTALB2Z · pith_short_16: UDGTCHTALB2ZZZ2J · pith_short_8: UDGTCHTA

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/UDGTCHTALB2ZZZ2JGXAIMDUKJ2 \
  | 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: a0cd311e6058759ce74935c0860e8a4eb90604f63c9a83932c45f30f55563801

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "dc7c788601a733cae98f11f3bf689d1385679beec1f130357257906034309c56",
    "cross_cats_sorted": [
      "cs.NA",
      "math.CV",
      "math.NA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-04-24T21:08:39Z",
    "title_canon_sha256": "9a2538a7f7face3e6d57549dedfb8566c883cc8b12960455a007a696ed0b812a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.23017",
    "kind": "arxiv",
    "version": 2
  }
}