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

arXiv: https://arxiv.org/abs/2606.17665 · HTML: https://mlsanity.com/paper/2606.17665/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 22 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [The landscape of the spiked tensor model](https://mlsanity.com/paper/1711.05424/) (2017)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [Stochastic gradient descent in high dimensions for multi-spiked tensor PCA](https://mlsanity.com/paper/2410.18162/) (2024)
- [Langevin dynamics for high-dimensional optimization: the case of multi-spiked tensor PCA](https://mlsanity.com/paper/2408.06401/) (2024)
- [Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test](https://mlsanity.com/paper/2406.18397/) (2024)

## Abstract (arXiv)

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical k-spin model. For Tensor PCA, we study 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 at signal-to-noise ratio~λ, through the profile maximum likelihood estimator, the MLE restricted to normalized rank-R tensors of coherence at least~kappa. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-(k,d) error bound recovers the asymptotically optimal rate~√dlog k of Perry, Wein and Bandeira, with explicit dependence on the rank~R and the coherence~kappa. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical k-spin Hamiltonian recovers the Auffinger--Ben Arous--vCern'y complexity function in the high-dimensional limit.