Companion code for the paper
Non-asymptotic Tail Bounds for the Kostlan–Shub–Smale Field: Tensor PCA and Spherical k-Spin Complexity Jean-Marc Azaïs, Federico Dalmao, Yohann De Castro.
The paper bounds the tail of the supremum of the Kostlan–Shub–Smale random field on the sphere through a Kac–Rice reduction to a shifted Gaussian Orthogonal Ensemble, and uses it to obtain (i) a non-asymptotic error bound for the profile maximum-likelihood estimator in spiked Tensor PCA, and (ii) a non-asymptotic two-sided bracketing of the annealed complexity of the spherical k-spin Hamiltonian. This repository contains the two notebooks that certify every constant and closed form in the paper and that reproduce its figures.
| File | Description |
|---|---|
Verification_Tensor_KSS_KacRice.ipynb |
Symbolic + numerical certification of the manuscript. Re-derives every load-bearing identity, constant, closed form, and inequality. 450/450 checks pass across 14 sections — 430 symbolic (SymPy, proved exactly, no numeric fallback) and 20 adaptive quadrature (the closed-form δ_exact against the Kac–Rice integral, to ~10⁻⁸–10⁻¹⁶ relative error). |
figures_article.ipynb |
Reproduces the figures: the tail-bound comparison δ_exact/δ_IMF/δ_SMF/δ_SM vs. the baseline (delta_min_comparison.pdf), the two-sided LRT threshold inversion (threshold_inversion_panels.pdf), and a supplementary normalized-ratio plot (delta_ratio_baseline.pdf). |
LICENSE |
MIT License. |
- Python ≥ 3.10
numpy,scipy,sympy,matplotlib,jupyter
pip install numpy scipy sympy matplotlib jupyterOpen either notebook in Jupyter and run all cells, or execute headlessly:
jupyter nbconvert --to notebook --execute Verification_Tensor_KSS_KacRice.ipynb
jupyter nbconvert --to notebook --execute figures_article.ipynbA clean run of the verification notebook ends with
OVERALL: 450/450 checks passed across 14 sections.
symbolic (SymPy) : 430 passed
adaptive quadrature : 20 passed
The verification notebook covers, among others: the Mehta–Fyodorov representation and the GOE one-point (Fyodorov/Mehta) identities; the four-piece closed form of δ_exact and the Hermite-tail recurrences J_mᵝ, K_jᵝ; the IMF / SMF / SM bounds and their nesting; the prefactor asymptotics; the Ben Arous–Dembo–Guionnet spectral-radius lemma and the Step-4 amplifier constants (C₁=64, C₂=8, C_amp=8√2); the spherical k-spin threshold constants (ρE∞=√2, E_BDG) and the Auffinger–Ben Arous–Černý complexity match Θₖ.
@article{azais2026nonasymptotic,
title = {Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field:
Tensor PCA and Spherical $k$-Spin Complexity},
author = {Aza{\"\i}s, Jean-Marc and Dalmao, Federico and De Castro, Yohann},
year = {2026}
}A companion paper on the exact (t-)spacing test for the same model is Azaïs, Dalmao, De Castro, Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test, arXiv:2406.18397.
Released under the MIT License.
