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Empirical Bounds on Linear Regions of Deep Rectifier Networks

arXiv 1810.03370 ↗First posted 8 Oct 2018cs.LG cs.AI math.OCcited by 23

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Abstract (arXiv)

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a fast proxy for the number of linear regions of a deep neural network.

Key facts. Cited by 23 papers in the map (data through 7 Oct 2026). First posted on arXiv 8 Oct 2018.

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Citing papers per year: 2021: 5; 2022: 2; 2023: 7; 2024: 6; 2025: 3.

23 papers.

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  1. 2021Towards Lower Bounds on the Depth of ReLU Neural Networkscited by 55
  2. 2022The Combinatorial Brain Surgeon: Pruning Weights That Cancel One Another in Neural Networkscited by 41
  3. 2023When Deep Learning Meets Polyhedral Theory: A Surveycited by 39
  4. 2021Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sumscited by 37
  5. 2024Scalable Neural Network Verification with Branch-and-bound Inferred Cutting Planescited by 33
  6. 2021Scaling Up Exact Neural Network Compression by ReLU Stabilitycited by 19
  7. 2021On the Expected Complexity of Maxout Networkscited by 15
  8. 2023TransNet: Transferable Neural Networks for Partial Differential Equationscited by 13
  9. 2024Optimization Over Trained Neural Networks: Taking a Relaxing Walkcited by 10
  10. 2021Modeling Design and Control Problems Involving Neural Network Surrogatescited by 8
  11. 2023Getting Away with More Network Pruning: From Sparsity to Geometry and Linear Regionscited by 8
  12. 2023Any Deep ReLU Network is Shallowcited by 6
11 more
  1. 2023Computational Tradeoffs of Optimization-Based Bound Tightening in ReLU Networkscited by 6
  2. 2024Feed-Forward Neural Networks as a Mixed-Integer Programcited by 4
  3. 2022Training Thinner and Deeper Neural Networks: Jumpstart Regularizationcited by 3
  4. 2025Relating Piecewise Linear Kolmogorov Arnold Networks to ReLU Networkscited by 2
  5. 2024A Computational Tropical Geometry Framework for Neural Networkscited by 1
  6. 2023Instance-wise Linearization of Neural Network for Model Interpretationcited by 0
  7. 2023Improving the Expressive Power of Deep Neural Networks through Integral Activation Transformcited by 0
  8. 2024Parallel Algorithms for Exact Enumeration of Deep Neural Network Activation Regionscited by 0
  9. 2024Tightening convex relaxations of trained neural networks: a unified approach for convex and S-shaped activationscited by 0
  10. 2025ReLU Networks as Random Functions: Their Distribution in Probability Spacecited by 0
  11. 2025Optimization over Trained (and Sparse) Neural Networks: A Surrogate within a Surrogatecited by 0