Sheaf Cohomology of Linear Predictive Coding Networks
This provides diagnostic tools and design principles for recurrent predictive coding networks, addressing a specific theoretical bottleneck in this domain.
The paper tackles the problem of analyzing error patterns in linear predictive coding networks by reformulating them as cellular sheaves, revealing that sheaf cohomology characterizes irreducible errors and that feedback loops can cause internal contradictions that stall learning.
Predictive coding (PC) replaces global backpropagation with local optimization over weights and activations. We show that linear PC networks admit a natural formulation as cellular sheaves: the sheaf coboundary maps activations to edge-wise prediction errors, and PC inference is diffusion under the sheaf Laplacian. Sheaf cohomology then characterizes irreducible error patterns that inference cannot remove. We analyze recurrent topologies where feedback loops create internal contradictions, introducing prediction errors unrelated to supervision. Using a Hodge decomposition, we determine when these contradictions cause learning to stall. The sheaf formalism provides both diagnostic tools for identifying problematic network configurations and design principles for effective weight initialization for recurrent PC networks.