Why Deep Jacobian Spectra Separate: Depth-Induced Scaling and Singular-Vector Alignment

Abstract

Understanding why gradient-based training in deep networks exhibits strong implicit bias remains challenging, in part because tractable singular-value dynamics are typically available only for balanced deep linear models. We propose an alternative route based on two theoretically grounded and empirically testable signatures of deep Jacobians: depth-induced exponential scaling of ordered singular values and strong spectral separation. Adopting a fixed-gates view of piecewise-linear networks, where Jacobians reduce to products of masked linear maps within a single activation region, we prove the existence of Lyapunov exponents governing the top singular values at initialization, give closed-form expressions in a tractable masked model, and quantify finite-depth corrections. We further show that sufficiently strong separation forces singular-vector alignment in matrix products, yielding an approximately shared singular basis for intermediate Jacobians. Together, these results motivate an approximation regime in which singular-value dynamics become effectively decoupled, mirroring classical balanced deep-linear analyses without requiring balancing. Experiments in fixed-gates settings validate the predicted scaling, alignment, and resulting dynamics, supporting a mechanistic account of emergent low-rank Jacobian structure as a driver of implicit bias.