Neural Networks as Local-to-Global Computations
This provides a novel mathematical framework for analyzing and training neural networks, potentially enabling local training without backpropagation, but it is incremental as it builds on existing sheaf theory and is validated only on small synthetic tasks.
The authors tackled the problem of interpreting neural networks by constructing a cellular sheaf from feedforward ReLU networks, showing that the forward pass output is the unique harmonic extension of boundary data and enabling bidirectional information propagation via a heat equation. They validated this on synthetic tasks, confirming convergence and scaling laws, though sheaf-based training was not yet competitive with SGD.
We construct a cellular sheaf from any feedforward ReLU neural network by placing one vertex for each intermediate quantity in the forward pass and encoding each computational step - affine transformation, activation, output - as a restriction map on an edge. The restricted coboundary operator on the free coordinates is unitriangular, so its determinant is $1$ and the restricted Laplacian is positive definite for every activation pattern. It follows that the relative cohomology vanishes and the forward pass output is the unique harmonic extension of the boundary data. The sheaf heat equation converges exponentially to this output despite the state-dependent switching introduced by piecewise linear activations. Unlike the forward pass, the heat equation propagates information bidirectionally across layers, enabling pinned neurons that impose constraints in both directions, training through local discrepancy minimization without a backward pass, and per-edge diagnostics that decompose network behavior by layer and operation type. We validate the framework experimentally on small synthetic tasks, confirming the convergence theorems and demonstrating that sheaf-based training, while not yet competitive with stochastic gradient descent, obeys quantitative scaling laws predicted by the theory.