Sheaf Cohomology of Linear Predictive Coding Networks

Jeffrey Seely
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:478-491, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-seely26a, title = {Sheaf Cohomology of Linear Predictive Coding Networks}, author = {Seely, Jeffrey}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {478--491}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/seely26a/seely26a.pdf}, url = {https://proceedings.mlr.press/v282/seely26a.html}, 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.} }
Endnote
%0 Conference Paper %T Sheaf Cohomology of Linear Predictive Coding Networks %A Jeffrey Seely %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-seely26a %I PMLR %P 478--491 %U https://proceedings.mlr.press/v282/seely26a.html %V 282 %X 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.
APA
Seely, J.. (2026). Sheaf Cohomology of Linear Predictive Coding Networks. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:478-491 Available from https://proceedings.mlr.press/v282/seely26a.html.

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