mlsanityData to 7 Oct 2026

The Geometry of Deep Networks: Power Diagram Subdivision

arXiv 1905.08443 ↗First posted 21 May 2019cs.LG stat.MLcited by 53

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Abstract (arXiv)

We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be em max-affine spline operators (MASOs) that partition their input space and apply a region-dependent affine mapping to their input to produce their output. We demonstrate that each MASO layer's input space partitioning corresponds to a em power diagram (an extension of the classical Voronoi tiling) with a number of regions that grows exponentially with respect to the number of units (neurons). We further show that a composition of MASO layers (e.g., the entire DN) produces a progressively subdivided power diagram and provide its analytical form. The subdivision process constrains the affine maps on the (exponentially many) power diagram regions to greatly reduce their complexity. For classification problems, we obtain a formula for a MASO DN's decision boundary in the input space plus a measure of its curvature that depends on the DN's nonlinearities, weights, and architecture. Numerous numerical experiments support and extend our theoretical results.

Key facts. Cited by 53 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 May 2019.

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Citing papers per year: 2021: 9; 2022: 15; 2023: 6; 2024: 8; 2025: 5; 2026: 10.

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  1. 2021High Fidelity Visualization of What Your Self-Supervised Representation Knows Aboutcited by 64
  2. 2021Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sumscited by 37
  3. 2022Interpreting Neural Networks through the Polytope Lenscited by 33
  4. 2023SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundariescited by 31
  5. 2021MaGNET: Uniform Sampling from Deep Generative Network Manifolds Without Retrainingcited by 23
  6. 2024The Real Tropical Geometry of Neural Networkscited by 23
  7. 2022On the Number of Regions of Piecewise Linear Neural Networkscited by 21
  8. 2022Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networkscited by 21
  9. 2022POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networkscited by 19
  10. 2022Toward a Geometrical Understanding of Self-supervised Contrastive Learningcited by 17
  11. 2021On the Expected Complexity of Maxout Networkscited by 15
  12. 2023Polyhedral Complex Extraction from ReLU Networks using Edge Subdivisioncited by 14
38 more
  1. 2022Adversarial robustness of sparse local Lipschitz predictorscited by 11
  2. 2023Characterizing Large Language Model Geometry Helps Solve Toxicity Detection and Generationcited by 11
  3. 2022Origami in N dimensions: How feed-forward networks manufacture linear separabilitycited by 9
  4. 2022Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networkscited by 9
  5. 2022Batch Normalization Explainedcited by 8
  6. 2021Max-Affine Spline Insights Into Deep Network Pruningcited by 5
  7. 2022What does a deep neural network confidently perceive? The effective dimension of high certainty class manifolds and their low confidence boundariescited by 5
  8. 2024Fast and Exact Enumeration of Deep Networks Partitions Regionscited by 5
  9. 2026Most ReLU Networks Admit Identifiable Parameterscited by 5
  10. 2021Fast Jacobian-Vector Product for Deep Networkscited by 4
  11. 2025Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Timecited by 4
  12. 2022Few-shot Learning as Cluster-induced Voronoi Diagrams: A Geometric Approachcited by 3
  13. 2022Analysis of function approximation and stability of general DNNs in directed acyclic graphs using un-rectifying analysiscited by 3
  14. 2023Training Dynamics of Deep Network Linear Regionscited by 3
  15. 2026The Linear Centroids Hypothesis: Features as Directions Learned by Local Expertscited by 3
  16. 2021Max-Linear Regression by Convex Programmingcited by 2
  17. 2022Spatial Transformer K-Meanscited by 2
  18. 2024A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on n-dimensional Classescited by 2
  19. 2024TTVD: Towards a Geometric Framework for Test-Time Adaptation Based on Voronoi Diagramcited by 2
  20. 2025GrokAlign: Geometric Characterisation and Acceleration of Grokkingcited by 2
  21. 2023Representation and decomposition of functions in DAG-DNNs and structural network pruningcited by 1
  22. 2024A rank decomposition for the topological classification of neural representationscited by 1
  23. 2026Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networkscited by 1
  24. 2026AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networkscited by 1
  25. 2026Characterizing the Discrete Geometry of ReLU Networkscited by 1
  26. 2021Learning DNN networks using un-rectifying ReLU with compressed sensing applicationcited by 0
  27. 2021Extracting Global Dynamics of Loss Landscape in Deep Learning Modelscited by 0
  28. 2022Progressive Voronoi Diagram Subdivision: Towards A Holistic Geometric Framework for Exemplar-free Class-Incremental Learningcited by 0
  29. 2022Enumeration of max-pooling responses with generalized permutohedracited by 0
  30. 2023CQnet: convex-geometric interpretation and constraining neural-network trajectoriescited by 0
  31. 2024On the Geometry of Deep Learningcited by 0
  32. 2024GleanVec: Accelerating vector search with minimalist nonlinear dimensionality reductioncited by 0
  33. 2024Combinatorial Regularity for Relatively Perfect Discrete Morse Gradient Vector Fields of ReLU Neural Networkscited by 0
  34. 2025A Quotient Homology Theory of Representation in Neural Networkscited by 0
  35. 2025Fitting Generalized Power Diagrams to 3D Image Data: A Prerequisite for Virtual Materials Testingcited by 0
  36. 2025Non-Singularity of the Gradient Descent map for Neural Networks with Piecewise Analytic Activationscited by 0
  37. 2026PLATE: Plasticity-Tunable Efficient Adapters for Geometry-Aware Continual Learningcited by 0
  38. 2026The Geometric Structure of Models Learning Sparse Datacited by 0

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