The Geometry of Limits in ReLU Function Spaces
Read the original on arXiv Machine Learning →The paper investigates the geometry and closure properties of function spaces realized by ReLU neural networks with fixed architecture. It proves that the minimal number of convex pieces in continuous piecewise-linear functions is lower semicontinuous under pointwise convergence, and uses this to show that shallow scalar ReLU realization spaces are closed in the weighted supremum norm, while providing a counterexample that demonstrates non-closedness for a specific architecture. The work also characterizes the role of a blow‑up residue in non‑closedness and offers new geometric and functional‑analytic tools for analyzing ReLU realization spaces.
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