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The Computational Complexity of Counting Linear Regions in ReLU Neural Networks
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Systematizes six definitions of linear regions in ReLU networks and analyzes the complexity of counting them, proving NP/#P-hardness already for shallow nets, inapproximability for deeper nets, and polynomial-space counting for some definitions.
Problem
Counting linear regions of ReLU networks is widely used as an expressivity measure, but the definitions of a linear region are inconsistent across papers and the computational complexity of counting them is largely unknown.
Approach
Extracts six non-equivalent definitions of linear regions, relates them via inequalities, and gives tight complexity classifications per definition: polynomial-time decidability for one hidden layer, #P-hardness for counting, NP-hardness and inapproximability for deeper networks, and polynomial-space counting algorithms.
Key results
- Deciding whether a one-hidden-layer ReLU network has more than one linear region is in P for all six definitions, while exact counting is #P-hard (and #P-complete for some definitions).
- For any fixed K and L>=2 hidden layers, K-region-decision is NP-hard; for L>=3 hidden layers, approximating linear region counting within ratio larger than (2^n+1)^-1 is NP-hard.
- Counting linear regions is in polynomial space for three of the six definitions.
Surprising findings
- Deciding whether a one-hidden-layer network has more than one linear region is polynomial-time, yet exact counting is #P-hard.
- A prior claim (Lemma 11(d) in an earlier paper) and an algorithm for closed connected regions were found to be inaccurate under the stated definitions.
Limitations
- Approximation hardness for one hidden layer and polynomial-factor approximation remain open.
- Counting for the closed connected region definition in #P membership is open.
Experimental setup
Builds on: hyperplane arrangement cell counting Serra et al. 2018 counting linear regions via MIP/ hardness of deep network regi
Closest papers by meaning
- Comparing Linear Regions in ReLU-Type Networks: Theory and Monte Carlo Methods (NeurIPS 2026 · similarity 0.93)
- Bounding and Counting Linear Regions of Deep Neural Networks (ICML 2018 · similarity 0.92)
- On the Local Complexity of Linear Regions in Deep ReLU Networks (ICML 2025 · similarity 0.90)
- Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training (ICML 2025 · similarity 0.88)
- Depth-Bounds for Neural Networks via the Braid Arrangement (NeurIPS 2025 · similarity 0.86)
- On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth (ICLR 2025 · similarity 0.86)
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