Pith Number

pith:GIDDWI6G

pith:2026:GIDDWI6GPXZGBOP4XNF2RZ4K6F

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Fast and Stable Gradient Approximation for Bilinear Forms of Hermitian Matrix Functions

Kipton Barros, Navjot Singh, Xiaoye Sherry Li

A forward-only gradient approximation reuses the Lanczos pass to stably differentiate bilinear forms u^T f(A(θ))v for Hermitian A with error proportional to the residual norm.

arxiv:2605.12801 v1 · 2026-05-12 · math.NA · cs.NA

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Claims

C1strongest claim

We propose a forward-only gradient approximation that reuses the Lanczos pass and adds very minimal overhead in most cases. We prove that its error is proportional to the Lanczos residual norm, the same quantity controlling the forward approximation. Whereas a traditional adjoint-based calculation would be unstable without reorthogonalization, the new method appears unconditionally stable in our tests.

C2weakest assumption

The approach assumes A(θ) remains Hermitian so that Lanczos produces real eigenvalues and orthogonal vectors, and that the residual norm from the forward Lanczos pass is a reliable error indicator for both the value and the gradient; stability is reported from tests but the abstract does not specify the range of matrices or functions f where this holds without additional conditions.

C3one line summary

A forward-only Lanczos gradient approximation for Hermitian matrix function bilinear forms whose error scales with the same residual norm as the forward approximation and appears stable without reorthogonalization.

References

42 extracted · 42 resolved · 0 Pith anchors

[1] Al-Mohy and Nicholas J 2009 · doi:10.1137/080716426

[2] Learning quantum systems.Nature Reviews Physics, 5:141–156, 2023 2023

[3] Error estimates and evaluation of matrix functions via the Faber transform.SIAM Journal on Numerical Analysis, 47(5):3849–3883, 2009 2009 · doi:10.1137/080741744

[4] Matrix functions in network analysis.GAMM-Mitteilungen, 43 (3):e202000012, 2020 2020

[5] Krylov-aware stochastic trace estimation.SIAM Journal on Matrix Analysis and Applications, 44(3):1218–1244, 2023 2023 · doi:10.1137/22m1526357

Formal links

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Receipt and verification

First computed	2026-05-18T03:09:12.715358Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

32063b23c67df260b9fcbb4ba8e78af17deaeb8535a326a7f0cc3118cdb1073e

Aliases

arxiv: 2605.12801 · arxiv_version: 2605.12801v1 · doi: 10.48550/arxiv.2605.12801 · pith_short_12: GIDDWI6GPXZG · pith_short_16: GIDDWI6GPXZGBOP4 · pith_short_8: GIDDWI6G

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/GIDDWI6GPXZGBOP4XNF2RZ4K6F \
  | 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: 32063b23c67df260b9fcbb4ba8e78af17deaeb8535a326a7f0cc3118cdb1073e

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "04f207fc71768aa24fb192bf2342fc9bc85e8507f3a4adae1714e3da8306e9ad",
    "cross_cats_sorted": [
      "cs.NA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NA",
    "submitted_at": "2026-05-12T22:40:16Z",
    "title_canon_sha256": "0e39afe8acce9c0af3e4e100532a4626c408b466cb73c24dbd593c3c15da7b1d"
  },
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  "source": {
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    "kind": "arxiv",
    "version": 1
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}