Binary perceptrons capacity via fully lifted random duality theory

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 fully lifted random duality theory (fl RDT) established in citeStojnicflrdt23 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 stationarized full lifting. The obtained results exactly match the replica symmetry breaking predictions obtained through statistical physics replica methods in citeKraMez89. Most notably, for the famous zero-threshold scenario, kappa=0, we uncover the well known α≈0.8330786 scaled capacity.