mlsanityData to 7 Oct 2026

SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries

arXiv 2302.12828 ↗First posted 24 Feb 2023cs.CV cs.LGcited by 31

Not on the map. Why?

Abstract (arXiv)

Current Deep Network (DN) visualization and interpretability methods rely heavily on data space visualizations such as scoring which dimensions of the data are responsible for their associated prediction or generating new data features or samples that best match a given DN unit or representation. In this paper, we go one step further by developing the first provably exact method for computing the geometry of a DN's mapping - including its decision boundary - over a specified region of the data space. By leveraging the theory of Continuous Piece-Wise Linear (CPWL) spline DNs, SplineCam exactly computes a DNs geometry without resorting to approximations such as sampling or architecture simplification. SplineCam applies to any DN architecture based on CPWL nonlinearities, including (leaky-)ReLU, absolute value, maxout, and max-pooling and can also be applied to regression DNs such as implicit neural representations. Beyond decision boundary visualization and characterization, SplineCam enables one to compare architectures, measure generalizability and sample from the decision boundary on or off the manifold. Project Website: bit.ly/splinecam.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). Builds on 24 of its 44 references that are in the map. First posted on arXiv 24 Feb 2023.

Builds on

24 of 44 references.

References · 24

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2019Analyzing and Improving the Image Quality of StyleGANcited by 3,580
  3. 2017Interpretation of Neural Networks is Fragilecited by 443
  4. 2014On the Number of Linear Regions of Deep Neural Networkscited by 378
  5. 2015Understanding Neural Networks Through Deep Visualizationcited by 332
  6. 2016On the Expressive Power of Deep Neural Networkscited by 302
  7. 2020Neural Architecture Search without Trainingcited by 241
  8. 2021Neural Architecture Search on ImageNet in Four GPU Hours: A Theoretically Inspired Perspectivecited by 168
  9. 2019Complexity of Linear Regions in Deep Networkscited by 166
  10. 2019Implicit Semantic Data Augmentation for Deep Networkscited by 106
  11. 2021A Structured Dictionary Perspective on Implicit Neural Representationscited by 100
  12. 2018Data augmentation instead of explicit regularizationcited by 62
12 more
  1. 2022Can Neural Nets Learn the Same Model Twice? Investigating Reproducibility and Double Descent from the Decision Boundary Perspectivecited by 57
  2. 2019The Geometry of Deep Networks: Power Diagram Subdivisioncited by 53
  3. 2020Empirical Studies on the Properties of Linear Regions in Deep Neural Networkscited by 36
  4. 2021CoIL: Coordinate-based Internal Learning for Imaging Inverse Problemscited by 33
  5. 2021CAMERAS: Enhanced Resolution And Sanity preserving Class Activation Mapping for image saliencycited by 28
  6. 2019Provable Certificates for Adversarial Examples: Fitting a Ball in the Union of Polytopescited by 25
  7. 2021MaGNET: Uniform Sampling from Deep Generative Network Manifolds Without Retrainingcited by 23
  8. 2022Polarity Sampling: Quality and Diversity Control of Pre-Trained Generative Networks via Singular Valuescited by 21
  9. 2022A Data-Augmentation Is Worth A Thousand Samples: Exact Quantification From Analytical Augmented Sample Momentscited by 14
  10. 2022Are All Linear Regions Created Equal?cited by 13
  11. 2022Batch Normalization Explainedcited by 8
  12. 2017An Adaptive Strategy for Active Learning with Smooth Decision Boundarycited by 6

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 9; 2025: 8; 2026: 11.

31 papers.

Sort:

By area:

  1. 2024Deep Networks Always Grok and Here is Whycited by 49
  2. 2024The Real Tropical Geometry of Neural Networkscited by 23
  3. 2023Polyhedral Complex Extraction from ReLU Networks using Edge Subdivisioncited by 14
  4. 2024Learning Transferable Features for Implicit Neural Representationscited by 14
  5. 2024On the Local Complexity of Linear Regions in Deep ReLU Networkscited by 9
  6. 2024What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Modelscited by 7
  7. 2025Convexity in ReLU Neural Networks: beyond ICNNs?cited by 6
  8. 2023Deep ReLU Networks Have Surprisingly Simple Polytopescited by 5
  9. 2023Training Dynamics of Deep Network Linear Regionscited by 3
  10. 2025Marching Neurons: Accurate Surface Extraction for Neural Implicit Shapescited by 3
  11. 2026The Linear Centroids Hypothesis: Features as Directions Learned by Local Expertscited by 3
  12. 2024TTVD: Towards a Geometric Framework for Test-Time Adaptation Based on Voronoi Diagramcited by 2
19 more
  1. 2025GrokAlign: Geometric Characterisation and Acceleration of Grokkingcited by 2
  2. 2024Polyhedral Complex Derivation from Piecewise Trilinear Networkscited by 1
  3. 2024A Computational Tropical Geometry Framework for Neural Networkscited by 1
  4. 2025mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networkscited by 1
  5. 2025Guidelines For The Choice Of The Baseline in XAI Attribution Methodscited by 1
  6. 2025Feature Attribution from First Principlescited by 1
  7. 2026Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networkscited by 1
  8. 2026AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networkscited by 1
  9. 2026Region Seeding via Pre-Activation Regularization: A Geometric View of Piecewise Affine Neural Networkscited by 1
  10. 2026Axiomatizing Neural Networks via Pursuit of Subspacescited by 1
  11. 2024On the Geometry of Deep Learningcited by 0
  12. 2025Geometric Integration for Neural Control Variatescited by 0
  13. 2025Non-Singularity of the Gradient Descent map for Neural Networks with Piecewise Analytic Activationscited by 0
  14. 2026Complexity of Linear Regions in Self-supervised Deep ReLU Networkscited by 0
  15. 2026What Cohort INRs Encode and Where to Freeze Themcited by 0
  16. 2026The Geometric Structure of Models Learning Sparse Datacited by 0
  17. 2026Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexitycited by 0
  18. 2026Affine Geometry of Gaussian ReLU Networks via Conditional Kac-Rice Formulascited by 0
  19. 2026Dynamics to decision: A mathematical theory of Lyapunov spectra and decision boundaries in deep classifierscited by 0