Research map / Architectures and efficiency / CNN design
On the Number of Linear Regions of Convolutional Neural Networks
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's title only (no summary yet). Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a depth-L ReLU CNN instead of a shallow CNN with asymptotically the same parameter count | ReLU CNN · Fixed input dimension; equal-parameter deep-versus-shallow comparison | linear regions per parameter: Ω(1/L · floor(d/d0)^(d0 (L-1) n^2 -2)) vs O((Ld^2)^(d0 n^2 -1)) | Helped |
| Use a convolutional rather than fully-connected ReLU architecture | ReLU CNN and fully-connected ReLU NN · Asymptotically equal parameters, input dimension, and number of layers | linear regions: R_N1/R_N3 ≥ Ω((√(2π n^2d0))^L (n^2/e)^(L d0 n^2)) | Helped |
| Count convolution-induced dependent hyperplane arrangements using the proposed exact formula rather than the fully-connected NN upper bound | One-layer CNN example · Input 1×3×1; filters 1×2×1; stride 1; d1=3 | linear regions: 40 vs 42 | No clear effect |
| Estimate region counts by sampling input points | Two-layer CNN example · Input 1×4×1; first layer 2 filters; second-layer d2 values | linear regions: 170, 261, 685, 1186, 1796, 2725, 3398, 4822 | Mixed |
| Increase the number of filters in a one-layer CNN | One-layer ReLU CNN · Fixed input and filter geometry | maximal linear regions: R_N = Θ(d1^#∪S_i,j) | Helped |
Closest papers by meaning
- Complexity of Linear Regions in Deep Networks (ICML 2019 · similarity 0.92)
- The Computational Complexity of Counting Linear Regions in ReLU Neural Networks (NeurIPS 2025 · similarity 0.90)
- Bounding and Counting Linear Regions of Deep Neural Networks (ICML 2018 · similarity 0.90)
- Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training (ICML 2025 · similarity 0.86)
- On the Local Complexity of Linear Regions in Deep ReLU Networks (ICML 2025 · similarity 0.86)
- From Alexnet to Transformers: Measuring the Non-linearity of Deep Neural Networks with Affine Optimal Transport (CVPR 2025 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.