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Tropical Geometry of Deep Neural Networks

arXiv 1805.07091 ↗First posted 18 May 2018cs.LG math.AG stat.MLcited by 92

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Abstract (arXiv)

We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions. An insight from our tropical formulation is that a deeper network is exponentially more expressive than a shallow network.

Key facts. Cited by 92 papers in the map (data through 7 Oct 2026). First posted on arXiv 18 May 2018.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 9; 2022: 11; 2023: 17; 2024: 17; 2025: 19; 2026: 19.

92 papers.

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  1. 2021Towards Lower Bounds on the Depth of ReLU Neural Networkscited by 55
  2. 2023When Deep Learning Meets Polyhedral Theory: A Surveycited by 39
  3. 2021Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sumscited by 37
  4. 2023Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopescited by 36
  5. 2024The Real Tropical Geometry of Neural Networkscited by 23
  6. 2021An enriched category theory of language: from syntax to semanticscited by 21
  7. 2022Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networkscited by 21
  8. 2025Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometrycited by 21
  9. 2022Functional dimension of feedforward ReLU neural networkscited by 20
  10. 2022Neural Networks are Decision Treescited by 16
  11. 2023Mildly Overparameterized ReLU Networks Have a Favorable Loss Landscapecited by 16
  12. 2021On the Expected Complexity of Maxout Networkscited by 15
38 more
  1. 2022Training Fully Connected Neural Networks is existsℝ-Completecited by 15
  2. 2024Activation degree thresholds and expressiveness of polynomial neural networkscited by 14
  3. 2021Neurons on Amoebaecited by 11
  4. 2023Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classificationcited by 11
  5. 2024On Minimal Depth in Neural Networkscited by 11
  6. 2025Depth-Bounds for Neural Networks via the Braid Arrangementcited by 11
  7. 2025On the Depth of Monotone ReLU Neural Networks and ICNNscited by 11
  8. 2025Better Neural Network Expressivity: Subdividing the Simplexcited by 11
  9. 2022Origami in N dimensions: How feed-forward networks manufacture linear separabilitycited by 9
  10. 2022Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networkscited by 9
  11. 2023Abstract Visual Reasoning: An Algebraic Approach for Solving Raven's Progressive Matricescited by 9
  12. 2024Decomposition Polyhedra of Piecewise Linear Functionscited by 9
  13. 2024On the Local Complexity of Linear Regions in Deep ReLU Networkscited by 9
  14. 2021Min-Max-Plus Neural Networkscited by 8
  15. 2024Neural Networks and (Virtual) Extended Formulationscited by 8
  16. 2023Noncompact uniform universal approximationcited by 7
  17. 2021Static analysis of ReLU neural networks with tropical polyhedracited by 5
  18. 2025The Resurrection of the ReLUcited by 5
  19. 2026Most ReLU Networks Admit Identifiable Parameterscited by 5
  20. 2023The Geometric Structure of Fully-Connected ReLU Layerscited by 4
  21. 2022MorphoActivation: Generalizing ReLU activation function by mathematical morphologycited by 3
  22. 2022A Combinatorial Perspective on the Optimization of Shallow ReLU Networkscited by 3
  23. 2023Synthesizing Programmatic Policies with Actor-Critic Algorithms and ReLU Networkscited by 3
  24. 2024TropNNC: Structured Neural Network Compression Using Tropical Geometrycited by 3
  25. 2025Constraining the outputs of ReLU neural networkscited by 3
  26. 2026Sparsity is Combinatorial Depth: Quantifying MoE Expressivity via Tropical Geometrycited by 3
  27. 2021On the space of coefficients of a Feed Forward Neural Networkcited by 2
  28. 2022FastSTMF: Efficient tropical matrix factorization algorithm for sparse datacited by 2
  29. 2023Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networkscited by 2
  30. 2024Complexity of Injectivity and Verification of ReLU Neural Networkscited by 2
  31. 2025Toric geometry of ReLU neural networkscited by 2
  32. 2026The Geometry of Thought: Disclosing the Transformer as a Tropical Polynomial Circuitcited by 2
  33. 2022Concentration inequalities and optimal number of layers for stochastic deep neural networkscited by 1
  34. 2023Predicting the cardinality and maximum degree of a reduced Gr"obner basiscited by 1
  35. 2023Matrix tri-factorization over the tropical semiringcited by 1
  36. 2023Tight Certified Robustness via Min-Max Representations of ReLU Neural Networkscited by 1
  37. 2024Tropical Decision Boundaries for Neural Networks Are Robust Against Adversarial Attackscited by 1
  38. 2024Polyhedral Complex Derivation from Piecewise Trilinear Networkscited by 1

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