We consider the classical \emph{spherical} perceptrons and study their capacities. The famous zero-threshold case was solved in the sixties of the last century (see, \cite{Wendel62,Winder,Cover65}) through the high-dimensional combinatorial considerations. The general threshold, $\kappa$, case though turned out to be much harder and stayed out of reach for the following several decades. A substantial progress was then made in \cite{SchTir02} and \cite{StojnicGardGen13} where the \emph{positive} threshold ($\kappa\geq 0$) scenario was finally fully settled. While the negative counterpart ($\kappa\leq 0$) remained out of reach, \cite{StojnicGardGen13} did show that the random duality theory (RDT) is still powerful enough to provide excellent upper bounds. Moreover, in \cite{StojnicGardSphNeg13}, a \emph{partially lifted} RDT variant was considered and it was shown that the upper bounds of \cite{StojnicGardGen13} can be lowered. After recent breakthroughs in studying bilinearly indexed (bli) random processes in \cite{Stojnicsflgscompyx23,Stojnicnflgscompyx23}, \emph{fully lifted} random duality theory (fl RDT) was developed in \cite{Stojnicflrdt23}. We here first show that the \emph{negative spherical perceptrons} can be fitted into the frame of the fl RDT and then employ the whole fl RDT machinery to characterize the capacity. To be fully practically operational, the fl RDT requires a substantial numerical work. We, however, uncover remarkable closed form analytical relations among key lifting parameters. Such a discovery enables performing the needed numerical calculations to obtain concrete capacity values. We also observe that an excellent convergence (with the relative improvement $\sim 0.1\%$) is achieved already on the third (second non-trivial) level of the \emph{stationarized} full lifting.

翻译：我们考虑经典的球面感知机并研究其容量。著名的零阈值情况在上世纪六十年代通过高维组合学考量得以解决（参见文献\cite{Wendel62,Winder,Cover65}）。然而，一般阈值$\kappa$的情况要困难得多，并在随后的几十年中一直无法攻克。随后在文献\cite{SchTir02}和\cite{StojnicGardGen13}中取得了重大进展，其中正阈值（$\kappa\geq 0$）场景最终被完全解决。尽管负阈值（$\kappa\leq 0$）场景仍无法触及，但\cite{StojnicGardGen13}确实表明随机对偶理论（RDT）仍足够强大以提供优异的上界。此外，在\cite{StojnicGardSphNeg13}中，考虑了一种部分提升的RDT变体，并证明了\cite{StojnicGardGen13}中的上界可以降低。继近期在文献\cite{Stojnicsflgscompyx23,Stojnicnflgscompyx23}中对双线性索引随机过程研究的突破之后，\cite{Stojnicflrdt23}中发展出了全提升随机对偶理论（fl RDT）。我们首先证明负球面感知机可以纳入fl RDT框架，然后利用完整的fl RDT机制来刻画其容量。为了实现充分的实践可操作性，fl RDT需要大量的数值计算。然而，我们发现了关键提升参数之间显著的闭式解析关系。这一发现使得能够执行所需的数值计算以获取具体的容量值。我们还观察到，在平稳化全提升的第三层（第二个非平凡层）上已实现了极佳的收敛性（相对改进幅度约$0.1\%$）。