# The Geometry of Deep Networks: Power Diagram Subdivision

arXiv: https://arxiv.org/abs/1905.08443 · HTML: https://mlsanity.com/paper/1905.08443/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2019.

## Built on by (most cited first, 30 of 53)

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- [SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries](https://mlsanity.com/paper/2302.12828/) (2023)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [MaGNET: Uniform Sampling from Deep Generative Network Manifolds Without Retraining](https://mlsanity.com/paper/2110.08009/) (2021)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [On the Number of Regions of Piecewise Linear Neural Networks](https://mlsanity.com/paper/2206.08615/) (2022)
- [POLICE: Provably Optimal Linear Constraint Enforcement for Deep Neural Networks](https://mlsanity.com/paper/2211.01340/) (2022)
- [Toward a Geometrical Understanding of Self-supervised Contrastive Learning](https://mlsanity.com/paper/2205.06926/) (2022)
- [On the Expected Complexity of Maxout Networks](https://mlsanity.com/paper/2107.00379/) (2021)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [Characterizing Large Language Model Geometry Helps Solve Toxicity Detection and Generation](https://mlsanity.com/paper/2312.01648/) (2023)
- [Adversarial robustness of sparse local Lipschitz predictors](https://mlsanity.com/paper/2202.13216/) (2022)
- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks](https://mlsanity.com/paper/2206.00228/) (2022)
- [Batch Normalization Explained](https://mlsanity.com/paper/2209.14778/) (2022)
- [Most ReLU Networks Admit Identifiable Parameters](https://mlsanity.com/paper/2605.03601/) (2026)
- [Max-Affine Spline Insights Into Deep Network Pruning](https://mlsanity.com/paper/2101.02338/) (2021)
- [What does a deep neural network confidently perceive? The effective dimension of high certainty class manifolds and their low confidence boundaries](https://mlsanity.com/paper/2210.05546/) (2022)
- [Fast and Exact Enumeration of Deep Networks Partitions Regions](https://mlsanity.com/paper/2401.11188/) (2024)
- [Fast Jacobian-Vector Product for Deep Networks](https://mlsanity.com/paper/2104.00219/) (2021)
- [Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Time](https://mlsanity.com/paper/2505.18023/) (2025)
- [Few-shot Learning as Cluster-induced Voronoi Diagrams: A Geometric Approach](https://mlsanity.com/paper/2202.02471/) (2022)
- [Analysis of function approximation and stability of general DNNs in directed acyclic graphs using un-rectifying analysis](https://mlsanity.com/paper/2206.05997/) (2022)
- [Training Dynamics of Deep Network Linear Regions](https://mlsanity.com/paper/2310.12977/) (2023)
- [The Linear Centroids Hypothesis: Features as Directions Learned by Local Experts](https://mlsanity.com/paper/2604.11962/) (2026)
- [Max-Linear Regression by Convex Programming](https://mlsanity.com/paper/2103.07020/) (2021)
- [Spatial Transformer K-Means](https://mlsanity.com/paper/2202.07829/) (2022)
- [A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on n-dimensional Classes](https://mlsanity.com/paper/2404.06104/) (2024)

## 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.