We study the statistical capacity of the classical binary perceptrons with general thresholds $\kappa$. After recognizing the connection between the capacity and the bilinearly indexed (bli) random processes, we utilize a recent progress in studying such processes to characterize the capacity. In particular, we rely on \emph{fully lifted} random duality theory (fl RDT) established in \cite{Stojnicflrdt23} to create a general framework for studying the perceptrons' capacities. Successful underlying numerical evaluations are required for the framework (and ultimately the entire fl RDT machinery) to become fully practically operational. We present results obtained in that directions and uncover that the capacity characterizations are achieved on the second (first non-trivial) level of \emph{stationarized} full lifting. The obtained results \emph{exactly} match the replica symmetry breaking predictions obtained through statistical physics replica methods in \cite{KraMez89}. Most notably, for the famous zero-threshold scenario, $\kappa=0$, we uncover the well known $\alpha\approx0.8330786$ scaled capacity.

翻译：我们研究具有一般阈值$\kappa$的经典二元感知器的统计容量。在认识到容量与双线性索引随机过程之间的联系后，我们利用该过程研究的最新进展来表征容量。特别地，我们依托文献\cite{Stojnicflrdt23}中建立的\emph{完全提升}随机对偶理论，构建了研究感知器容量的通用框架。该框架（及整个完全提升随机对偶理论机制）需要借助成功的数值评估才能真正投入实际应用。我们展示了这方面的研究成果，并揭示容量表征在\emph{平稳化}完全提升的第二层级（首个非平凡层级）上实现。所得结果\emph{精确}匹配文献\cite{KraMez89}中通过统计物理复制方法获得的复制对称破缺预测。最值得注意的是，对于著名的零阈值情形$\kappa=0$，我们获得了众所周知的缩比容量$\alpha\approx0.8330786$。