Spectral Concentration at the Edge of Stability: Information Geometry of Kernel Associative Memory

Recent advances using Kernel Logistic Regression (KLR) have demonstrated that learning can sculpt these landscapes to achieve capacities far exceeding classical limits [1-3]. Our previous phenomenological analysis identified a Ridge of Optimization where stability is maximized via a mechanism we termed Spectral Concentration, defined as a state where the weight spectrum exhibits a sharp hierarchy [4]. However, a deeper question remains: Why does the learning dynamics self-organize into this specific spectral state? Why does the system operate at the brink of instability? T o answer these questions, we must look beyond the Euclidean geometry of the weight parameters and consider the intrinsic geometry of the probability distributions they represent. This is the domain of Information Geometry [5]. In this work, we reinterpret the KLR Hopfield network as a statistical manifold equipped with a Fisher-Rao metric.