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

Eloi Martinet; Leon Bungert

cs.LGNov 6, 2026

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

Authors: Eloi Martinet, Leon Bungert

Abstract

We prove that the set $\mathcal{N}_N$ of ReLU networks with one hidden layer and at most $N\in\mathbb N$ neurons is weakly closed in $W^{1,p}(Ω)$ , for a bounded connected open set $Ω\subset\mathbb R^d$ . Consequently, every coercive, weakly lower semicontinuous energy on this space attains its minimum over $\mathcal{N}_N$ . This covers the Deep Ritz method with convex integrands, Sobolev training and Lipschitz learning. Moreover, as the weight decay parameter tends to zero, minimizers of the regularized problem converge, up to subsequences, to minimizers with minimal parameter norm. We also prove that the set of ReLU networks with two hidden layers of bounded widths is not closed in $W^{1,p}(Ω)$ or in $C(\overlineΩ)$ . This disproves a conjecture of Petersen, Raslan and Voigtlaender.

Figures & tables

Panel of Figure 1 — Figure 1: Construction of $x_{0}\in\Omega$ lying on $H_{\theta^{(k)}}$ but on no other limiting break hyperplane. Hence, on the ball $B(x_{0},r)$ the neural network $u^{n}$ for all $n\in\mathbb{N}$ large enough is of the form $u^{n}=v_{k}^{n}+\ell_{k}^{n}$ for $v_{k}^{n}$ as in 6 and some affine function $\ell_{k}^{n}\in\mathcal{A}$ .

Panel of Figure 2 — Figure 2: Definition of the set $D_{k,\varepsilon}$ on which $z_{\theta_{i}^{n}}$ with $i\in C_{k}$ and $z_{\theta^{(k)}}$ have the same sign for large enough $n\in\mathbb{N}$ since the former converges uniformly to the latter.

Panel of Figure 3 — Figure 3: A domain $\Omega$ with two connected components at positive distance from each other. The only hyperplane $H$ that separates the components is tangent to both of them, preventing $u$ to be represented by a neural network.

Panel of Figure 4 — Figure 4: The target $f$ on $Q$ with $\lambda=5$ for $d=2$ (left) and $d=3$ (right). For $d=3$ , $f$ does not depend on $x_{3}$ , and only the slices $\{x_{3}=0\}$ and $\{x_{3}=1\}$ are shown. The orange lines are the zero level sets $\{x_{2}-c_{j}-K(x)=0\}$ of the arguments of the $g_{j}$ (right side) and their mirror images $\{x_{2}-c_{j}-K(S(x))=0\}$ (left side); the dashed lines are the walls $\{x_{1}=k\}$ , $k=1,\dots,4$ . The set $\mathcal{P}$ used in the proof of Proposition 3.3 (iii) consists of twelve points for $d=2$ and of twelve segments parallel to the $x_{3}$ -axis for $d=3$ .

Figure 5: The networks $u^{\eta}\in\mathcal{N}_{4,6}$ with $\lambda=5$ for $\eta=0.4,\,0.2,\,0.1$ and their limit $f\in\mathcal{N}_{5,6}$ (bottom right), on $Q=(0,5)\times(0,2)$ .

Figure 6: The function

g(x)=\sigma(x_{2}-\kappa(x_{1}))

of Lemma 3.8 , for

\kappa

with a single breakpoint at

t_{0}

. The highlighted curve is

\{x_{2}=\kappa(x_{1})\}

, and

g

is not locally shallow at the point

P=(t_{0},\kappa(t_{0}))

. The kinks of

g

near

P

lie on three half-lines starting at

P

, whereas the kinks of a shallow network lie on full lines.

Figure 7: The counting argument in the proof of Proposition 3.3 (iii) , with

c_{j}=j/10

. Four walls contain the twelve points of

\mathcal{P}

(left), while a non-vertical first-layer line contains at most two points of

\mathcal{P}

(right), which leaves two points out.

ReLU Neural Networks

Rectified linear unit (ReLU) neural networks use piecewise-linear activations to study approximation, learning dynamics, expressivity, and verification.

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The Geometry of Limits in ReLU Function Spaces

Kathryn A. Lindsey

ReLU Neural Networks Neural Network Approximation Theory Neural Network Expressivity Shallow Neural Networks

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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.

Abstract

ReLU Neural Networks

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