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

• URL: http://arxiv.org/abs/2606.17665v1
• Date: Tue, 16 Jun 2026 08:27:25 GMT
• Title: Non-asymptotic Tail Bounds for the Kostlan--Shub--Smale Field: Tensor PCA and Spherical $k$-Spin Complexity
• Abstract summary: This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale random field on the sphere.<n>It applies it to two problems: Spiked PCA and the landscape of the spherical $k$-spin model.
• Score: 0.6627152091494143
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$λ$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$κ$. 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~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$κ$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

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