On the (non-)closedness of the set of ReLU networks

We study the geometry and closure of spaces of functions realized by ReLU neural networks of fixed architecture. We first establish a general result about limits of continuous piecewise-linear (CPWL) functions: the minimum number of convex sets on which a function is affine is lower semicontinuous under pointwise convergence. The proof uses a connection to hypergraph coloring and a compactness principle, and yields quantitative bounds on the polyhedral complexity of continuous limits. We then characterize the functions realizable by shallow ReLU networks with vector-valued outputs, extending a known scalar-valued characterization. Using this and the geometry of limiting hyperplane arrangements, we prove that fixed-width shallow scalar ReLU realization spaces are closed in the weighted supremum norm on the whole input space. In contrast, we construct an explicit sequence of networks with architecture $(2,3,2\mid1)$ whose limit is not realizable by that architecture. Non-closedness holds both globally in weighted supremum norm and under uniform convergence on compact sets with nonempty interior, disproving a conjecture of Petersen, Raslan, and Voigtlaender. Finally, we show that non-closedness for a fixed scalar-output ReLU architecture necessarily involves a blow-up residue, in which diverging output coefficients amplify an emerging linear dependence among hidden features. Together, these results provide geometric and functional-analytic tools for studying the boundaries and degenerations of ReLU realization spaces.

We study pathwise connectivity of sublevel sets for one-hidden-layer ReLU networks with constrained first-layer weights and an $\ell_1$ penalty on the output layer. The data term is assumed convex and globally Lipschitz in the scalar logit. We first give a finite-width construction that connects any two points of a common sublevel through a path controlled by a loss-consistent compression functional and a first-order perturbation term. The proof replaces the quadratic perturbation estimate in the Freeman--Bruna mechanism by a direct Lipschitz bound. Positive homogeneity is then used in a direction that is compatible with the penalty: every active atom is moved monotonically from the unit ball to the unit sphere while its output coefficient is reduced. Sphere covering and cluster merging consequently give $O(m^{-1/(n-1)})$ fixed-level thickening for $n\ge2$, while the one-dimensional two-ray dictionary gives exact connectivity for every $m\ge4$. We also prove internally that the regularized approximation values satisfy $e(l)-e_\infty=O(l^{-1/2})$. More generally, a rate $O(l^{-s})$ transfers to a near-optimal barrier rate $O(m^{-s/((n-1)s+1)})$; under the standing assumptions, this yields the explicit rate $O(m^{-1/(n+1)})$. A theorem-aligned finite-distribution experiment complements the analysis. The primary Huber run yields a maximal best certified upper gap $1.66\times10^{-5}$ over 720 recorded pairs at widths $m\ge16$; a matched binary-cross-entropy rerun and a 720-endpoint dense-representation stress test probe loss robustness and the active cluster-merging mechanism.

We construct a continuous function on $[-1,1]^2$ for which the infimum of the $L^2$-approximation error over the class of ReLU networks with two hidden layers and architecture $(2,2,1,1)$ is not attained. This counterexample shows that deep ReLU networks do not satisfy the best approximation property in general, even for continuous target functions. Our findings contrast with existing results for shallow neural networks, where ReLU is one of the few activation functions that guarantee the existence of global minima in typical regression tasks. The proof is based on the fact that the breaklines for neurons in the second hidden layer can bend, which breaks the symmetry arguments used in the one-hidden-layer case.