Sheaf Cohomology of Linear Predictive Coding Networks

TL;DR

This paper introduces a sheaf cohomology framework for linear predictive coding networks, providing new insights into their error patterns, feedback loop issues, and strategies for improved initialization.

Contribution

It formulates linear predictive coding networks as cellular sheaves, linking sheaf cohomology to error analysis and network diagnostics.

Findings

Sheaf cohomology characterizes irreducible error patterns.

Feedback loops can cause internal contradictions in recurrent networks.

Sheaf formalism aids in diagnosing and initializing recurrent PC networks.

Abstract

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.