Hierarchical Hopfield Network Decomposition: A Spiked Covariance Framework for Latent Prototype Discovery

We revisit the classical Hopfield network from a spiked covariance perspective, showing how the Hebbian coupling matrix forms a low-rank perturbation of the identity. This viewpoint links outlier eigenvalues in the sample covariance matrix to latent signal vectors, explaining how multiple signals can fuse into a single spurious state. We propose a hierarchical algorithm that uses Hopfield updates to iteratively partition the data, isolating more granular spiked subspaces until no further mergers remain. Unlike classical approaches focusing on capacity alone, our method reveals latent signals even when they are strongly correlated. Experiments on MNIST and LFW confirm that these signals serve as interpretable "prototypes" and improve clustering initialization.