# Max-Affine Spline Insights into Deep Generative Networks

arXiv: https://arxiv.org/abs/2002.11912 · HTML: https://mlsanity.com/paper/2002.11912/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2020.

## Built on by (most cited first, 10 of 10)

- [MaGNET: Uniform Sampling from Deep Generative Network Manifolds Without Retraining](https://mlsanity.com/paper/2110.08009/) (2021)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [What Secrets Do Your Manifolds Hold? Understanding the Local Geometry of Generative Models](https://mlsanity.com/paper/2408.08307/) (2024)
- [Fast Jacobian-Vector Product for Deep Networks](https://mlsanity.com/paper/2104.00219/) (2021)
- [Few-shot Learning as Cluster-induced Voronoi Diagrams: A Geometric Approach](https://mlsanity.com/paper/2202.02471/) (2022)
- [Max-Linear Regression by Convex Programming](https://mlsanity.com/paper/2103.07020/) (2021)
- [Rayleigh EigenDirections (REDs): GAN latent space traversals for multidimensional features](https://mlsanity.com/paper/2201.10423/) (2022)
- [Progressive Voronoi Diagram Subdivision: Towards A Holistic Geometric Framework for Exemplar-free Class-Incremental Learning](https://mlsanity.com/paper/2207.14202/) (2022)
- [The Data Manifold under the Microscope](https://mlsanity.com/paper/2606.15760/) (2026)
- [Laguerre Geometry for Interpreting Large Language Models](https://mlsanity.com/paper/2607.10578/) (2026)

## Abstract (arXiv)

We connect a large class of Generative Deep Networks (GDNs) with spline operators in order to derive their properties, limitations, and new opportunities. By characterizing the latent space partition, dimension and angularity of the generated manifold, we relate the manifold dimension and approximation error to the sample size. The manifold-per-region affine subspace defines a local coordinate basis; we provide necessary and sufficient conditions relating those basis vectors with disentanglement. We also derive the output probability density mapped onto the generated manifold in terms of the latent space density, which enables the computation of key statistics such as its Shannon entropy. This finding also enables the computation of the GDN likelihood, which provides a new mechanism for model comparison as well as providing a quality measure for (generated) samples under the learned distribution. We demonstrate how low entropy and/or multimodal distributions are not naturally modeled by DGNs and are a cause of training instabilities.