Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field: Tensor PCA and Spherical $k$-Spin Complexity

2026-06-17 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

The paper "Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field" (arXiv:2606.17665) introduces a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere. These bounds are applied to two complex problems: Spiked Tensor PCA and the landscape of the spherical k-spin model. For Tensor PCA, the authors analyze the non-asymptotic statistical limits of estimating a rank-R symmetric signal tensor of order k ≥ 3 and dimension d ≥ 3 from a single Gaussian observation. Their method uses a profile maximum likelihood estimator and a reduction via the Tube Method, bounding error by the KSS field's supremum. This yields a finite-(k,d) error bound that recovers the asymptotically optimal rate √dlog k of Perry, Wein and Bandeira, with explicit dependence on rank R and coherence κ. Additionally, the work provides a two-sided non-asymptotic bracketing of the annealed complexity for the spherical k-spin Hamiltonian, recovering the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

Key takeaway

For research scientists working on high-dimensional statistical inference or spin glass models, this work provides rigorous non-asymptotic tools. You can now use the explicit tail bounds for the Kostlan--Shub--Smale field to derive precise error guarantees for Spiked Tensor PCA, recovering optimal rates like √dlog k. This enables more accurate theoretical analysis and validation of estimators for complex tensor data and energy landscapes, offering a robust framework for understanding their statistical limits.

Key insights

Explicit non-asymptotic tail bounds for KSS fields enable precise analysis of high-dimensional statistical and spin glass models.

Principles

• KSS field supremum bounds are crucial for tensor estimation.
• Tube Method reduces complex estimation to KSS field analysis.
• Mehta--Fyodorov and Ben Arous--Dembo--Guionnet provide tail bounds.

Method

The method involves reducing estimation error to the KSS field supremum via the Tube Method and rank-reduction, then applying the Kac--Rice formula and a hierarchy of four explicit tail bounds derived from Mehta--Fyodorov and Ben Arous--Dembo--Guionnet representations.

In practice

• Apply KSS bounds to analyze Spiked Tensor PCA.
• Use bounds for spherical k-spin model complexity.
• Estimate rank-R tensors with profile MLE.

Topics

• Kostlan--Shub--Smale Field
• Non-asymptotic Tail Bounds
• Spiked Tensor PCA
• Spherical k-Spin Model
• Statistical Inference
• High-Dimensional Statistics

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.