Neural LoFi with Backward Coupling: A Spectral Theory of Cross-Layer Feature Learning

Yatin Dandi, Matteo Vilucchio, Hugo Tabanelli, Florent Krzakala

During joint training, backpropagation allows later layers to reshape earlier representations, a feedback mechanism absent from purely layer-wise learning. We extend Neural Low-Degree Filtering (Neural LoFi), a recently proposed layer-wise spectral surrogate for feature learning, to capture this interaction. Starting from the exact backpropagation gradients, we derive a local coupling between a neuron's incoming weights and its corresponding next-layer column. In a controlled small-initialization regime with smooth activations and fixed surroundings, these dynamics amplify singular modes of a cross-layer correlation operator. Each growing mode pairs an earlier feature with an upper-layer response through shared neuron coefficients. This motivates a backward-coupling algorithm in which upper-layer sensitivities generate sample-dependent effective labels for earlier spectral steps, while corrected features refine upward transmission. A function-space variational principle characterizes the selected feature-response pairs by maximizing the label correlation of their product under kernel-norm constraints. For three-layer networks, we derive self-consistent continuous-time dynamics…