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.

翻译：本文建立了球面上Kostlan–Shub–Smale（KSS）随机场上确界的显式非渐近尾部界层次结构，并将其应用于两个问题：尖峰张量PCA与球面$k$-自旋模型的景观。对于张量PCA，我们通过\textit{轮廓最大似然估计}（即限制在相干性至少为$κ$的归一化秩-$R$张量上的MLE）研究在信噪比$λ$下从单个高斯观测中估计阶$k\ge 3$、维度$d\ge 3$的秩-$R$对称信号张量的非渐近统计极限。我们的分析基于单一约化：一个确定性几何不等式（管方法）与秩约化步骤将估计误差界化为标准KSS场的上确界，而Kac–Rice公式将其转化为期望绝对特征多项式（关于平移高斯正交系综）的高斯积分，该多项式进一步由我们层次结构中的四个显式尾部界（三个来自Mehta–Fyodorov表示，一个来自Ben Arous–Dembo–Guionnet大偏差）控制。这一约化导出两个结果，均具有显式常数。对于估计，有限$(k,d)$误差界恢复了Perry、Wein和Bandeira的渐近最优速率$\sqrt{d\log k}$，并显式依赖于秩$R$和相干性$κ$。对于景观，球面$k$-自旋哈密顿量的退火复杂度的双边非渐近区间估计在高维极限下恢复了Auffinger–Ben Arous–Černý复杂度函数。