Self-Organization of Attractor Landscapes in High-Capacity Kernel Logistic Regression Hopfield Networks

Kernel-based learning methods can dramatically increase the storage capacity of Hopfield networks, yet the dynamical mechanism behind this enhancement remains poorly understood. We address this gap by conducting a geometric analysis of the network's energy landscape. We introduce a novel metric, ``Pinnacle Sharpness,'' to quantify the local stability of attractors. By systematically varying the kernel width and storage load, we uncover a rich phase diagram of attractor shapes. Our central finding is the emergence of a ``ridge of optimization,'' where the network maximizes attractor stability under challenging high-load and global-kernel conditions. Through a theoretical decomposition of the landscape gradient into a direct ``driving'' force and an indirect ``feedback'' force, we reveal the origin of this phenomenon. The optimization ridge corresponds to a regime of strong anti-correlation between the two forces, where the direct force, amplified by the high storage load, dominates the opposing collective feedback force. This demonstrates a sophisticated self-organization mechanism: the network adaptively harnesses inter-pattern interactions as a cooperative feedback control system to sculpt a robust energy landscape. Our findings provide a new physical picture for the stability of high-capacity associative memories and offer principles for their design.