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

Two separate research papers were submitted to arXiv on consecutive days in June 2026, one addressing non-asymptotic tail bounds for the Kostlan--Shub--Smale random field and the other focusing on additive manufacturing process optimization.

A paper submitted on 16 Jun 2026^[1] establishes a hierarchy of explicit, non-asymptotic tail bounds for the Kostlan--Shub--Smale random field on the sphere. The authors applied this hierarchy to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, the paper recovers the asymptotically optimal rate of $rac{ ext{d}}{ ext{2}} ext{log}( ext{k})$ for estimation, as previously reported by Perry, Wein, and Bandeira. Additionally, the paper recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit for the landscape of the spherical $k$-spin model. In a separate submission on 18 Jun 2026^[2], researchers presented a methodology for optimizing additive manufacturing processes, achieving a convergence value of 322.79 within 14 episodes. The authors highlighted that precise parameter control is crucial in minimizing defects such as porosity in additive manufacturing.