Rigorous justification for bounding the encoder Hessian via optimization dynamics

Establish rigorous theoretical conditions under which gradient-based optimization of a neural network encoder E yields a uniform bound on the Hessian spectral norm L_E = sup_{x∈M} ||∇^2E(x)||_2 across the data manifold M (or a compact subset), thereby providing a non-heuristic explanation for the empirically observed spectral bias that keeps L_E small and supports the Fisher Information Rate deviation guarantees for latent diffusion.

Background

The nonlinear stability analysis for Fisher Information Rate (FIR) requires a Taylor remainder bound that depends on the encoder’s uniform Hessian spectral norm L_E = sup_{x∈M} ||∇^2E(x)||_2. Proposition gpe-eps-delta shows that when L_E is small, together with controlled Jacobian distortion, the FIR deviation can be bounded for geometry-preserving encoders.

Empirically, neural network encoders trained by gradient-based methods exhibit spectral bias that suppresses high-frequency components, suggesting small second derivatives in practice. However, the paper notes that this reasoning is heuristic and lacks a rigorous theoretical justification linked to the optimization dynamics and architecture.

A formal proof that training dynamics enforce a uniform Hessian bound would strengthen the theoretical basis for FIR stability claims and clarify when latent spaces will maintain diffusability under learned encoders.