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

arXiv: https://arxiv.org/abs/2302.12828 · HTML: https://mlsanity.com/paper/2302.12828/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 24 Feb 2023.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Interpretation of Neural Networks is Fragile](https://mlsanity.com/paper/1710.10547/) (2017)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Understanding Neural Networks Through Deep Visualization](https://mlsanity.com/paper/1506.06579/) (2015)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Neural Architecture Search without Training](https://mlsanity.com/paper/2006.04647/) (2020)
- [Neural Architecture Search on ImageNet in Four GPU Hours: A Theoretically Inspired Perspective](https://mlsanity.com/paper/2102.11535/) (2021)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Implicit Semantic Data Augmentation for Deep Networks](https://mlsanity.com/paper/1909.12220/) (2019)
- [A Structured Dictionary Perspective on Implicit Neural Representations](https://mlsanity.com/paper/2112.01917/) (2021)
- [Data augmentation instead of explicit regularization](https://mlsanity.com/paper/1806.03852/) (2018)
- [Can Neural Nets Learn the Same Model Twice? Investigating Reproducibility and Double Descent from the Decision Boundary Perspective](https://mlsanity.com/paper/2203.08124/) (2022)
- [The Geometry of Deep Networks: Power Diagram Subdivision](https://mlsanity.com/paper/1905.08443/) (2019)
- [Empirical Studies on the Properties of Linear Regions in Deep Neural Networks](https://mlsanity.com/paper/2001.01072/) (2020)
- [CoIL: Coordinate-based Internal Learning for Imaging Inverse Problems](https://mlsanity.com/paper/2102.05181/) (2021)
- [CAMERAS: Enhanced Resolution And Sanity preserving Class Activation Mapping for image saliency](https://mlsanity.com/paper/2106.10649/) (2021)
- [Provable Certificates for Adversarial Examples: Fitting a Ball in the Union of Polytopes](https://mlsanity.com/paper/1903.08778/) (2019)
- [MaGNET: Uniform Sampling from Deep Generative Network Manifolds Without Retraining](https://mlsanity.com/paper/2110.08009/) (2021)
- [Polarity Sampling: Quality and Diversity Control of Pre-Trained Generative Networks via Singular Values](https://mlsanity.com/paper/2203.01993/) (2022)
- [A Data-Augmentation Is Worth A Thousand Samples: Exact Quantification From Analytical Augmented Sample Moments](https://mlsanity.com/paper/2202.08325/) (2022)
- [Are All Linear Regions Created Equal?](https://mlsanity.com/paper/2202.11749/) (2022)
- [Batch Normalization Explained](https://mlsanity.com/paper/2209.14778/) (2022)
- [An Adaptive Strategy for Active Learning with Smooth Decision Boundary](https://mlsanity.com/paper/1711.09294/) (2017)

## Built on by (most cited first, 30 of 31)

- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [Learning Transferable Features for Implicit Neural Representations](https://mlsanity.com/paper/2409.09566/) (2024)
- [On the Local Complexity of Linear Regions in Deep ReLU Networks](https://mlsanity.com/paper/2412.18283/) (2024)
- [What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Models](https://mlsanity.com/paper/2408.08307/) (2024)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Deep ReLU Networks Have Surprisingly Simple Polytopes](https://mlsanity.com/paper/2305.09145/) (2023)
- [Training Dynamics of Deep Network Linear Regions](https://mlsanity.com/paper/2310.12977/) (2023)
- [Marching Neurons: Accurate Surface Extraction for Neural Implicit Shapes](https://mlsanity.com/paper/2509.21007/) (2025)
- [The Linear Centroids Hypothesis: Features as Directions Learned by Local Experts](https://mlsanity.com/paper/2604.11962/) (2026)
- [TTVD: Towards a Geometric Framework for Test-Time Adaptation Based on Voronoi Diagram](https://mlsanity.com/paper/2412.07980/) (2024)
- [GrokAlign: Geometric Characterisation and Acceleration of Grokking](https://mlsanity.com/paper/2506.12284/) (2025)
- [Polyhedral Complex Derivation from Piecewise Trilinear Networks](https://mlsanity.com/paper/2402.10403/) (2024)
- [A Computational Tropical Geometry Framework for Neural Networks](https://mlsanity.com/paper/2405.20174/) (2024)
- [mPOLICE: Provable Enforcement of Multi-Region Affine Constraints in Deep Neural Networks](https://mlsanity.com/paper/2502.02434/) (2025)
- [Guidelines For The Choice Of The Baseline in XAI Attribution Methods](https://mlsanity.com/paper/2503.19813/) (2025)
- [Feature Attribution from First Principles](https://mlsanity.com/paper/2505.24729/) (2025)
- [Training-Time Batch Normalization Reshapes Local Partition Geometry in Piecewise-Affine Networks](https://mlsanity.com/paper/2605.04946/) (2026)
- [AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networks](https://mlsanity.com/paper/2605.06218/) (2026)
- [Region Seeding via Pre-Activation Regularization: A Geometric View of Piecewise Affine Neural Networks](https://mlsanity.com/paper/2605.06300/) (2026)
- [Axiomatizing Neural Networks via Pursuit of Subspaces](https://mlsanity.com/paper/2605.20534/) (2026)
- [On the Geometry of Deep Learning](https://mlsanity.com/paper/2408.04809/) (2024)
- [Geometric Integration for Neural Control Variates](https://mlsanity.com/paper/2509.15538/) (2025)
- [Non-Singularity of the Gradient Descent map for Neural Networks with Piecewise Analytic Activations](https://mlsanity.com/paper/2510.24466/) (2025)
- [Complexity of Linear Regions in Self-supervised Deep ReLU Networks](https://mlsanity.com/paper/2604.24393/) (2026)
- [What Cohort INRs Encode and Where to Freeze Them](https://mlsanity.com/paper/2605.08298/) (2026)
- [The Geometric Structure of Models Learning Sparse Data](https://mlsanity.com/paper/2605.08464/) (2026)
- [Expressivity Saturation: Reduced Affine Region Usage Under Increasing Task Complexity](https://mlsanity.com/paper/2606.21687/) (2026)
- [Affine Geometry of Gaussian ReLU Networks via Conditional Kac-Rice Formulas](https://mlsanity.com/paper/2609.32695/) (2026)

## 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.