New homology theory reveals topology of neural representations without geometry

A new paper from Kosio Beshkov proposes a quotient homology theory that extracts the intrinsic topology of neural network representations without relying on geometric assumptions. ReLU networks partition their input domain into convex polyhedra over which the network acts affinely. Beshkov leverages this to define an equivalence relation on input data, creating a quotient space split by the local ranks and intersections of these affine maps — the "overlap decomposition." He proves that if the intersections between each polyhedron and an input manifold are convex, the homology groups of the neural representation are isomorphic to the quotient homology groups of the manifold modulo this decomposition. This lets researchers calculate Betti numbers — topological invariants counting holes — directly from the network's structure, bypassing the need for a chosen metric.

The paper provides numerical algorithms via linear programming and a union-find data structure to compute the overlap decomposition. Experiments on simple datasets show that the resulting Betti numbers track purely topological features, unlike persistent homology which mixes geometry and topology. Beshkov also analyzes how the overlap decomposition evolves during training on classification tasks, revealing how networks learn topological structure. While promising, the method currently has limitations in scalability and convexity assumptions. The work offers a rigorous new lens for understanding what neural networks represent internally, with potential applications in interpretability and network analysis.