# DDSL: Deep Differentiable Simplex Layer for Learning Geometric Signals

arXiv: https://arxiv.org/abs/1901.11082 · HTML: https://mlsanity.com/paper/1901.11082/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2019.

## Built on by (most cited first, 5 of 5)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [TorchSpatial: A Location Encoding Framework and Benchmark for Spatial Representation Learning](https://mlsanity.com/paper/2406.15658/) (2024)
- [Towards General-Purpose Representation Learning of Polygonal Geometries](https://mlsanity.com/paper/2209.15458/) (2022)
- [Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications](https://mlsanity.com/paper/2408.14806/) (2024)
- [PolygonGNN: Representation Learning for Polygonal Geometries with Heterogeneous Visibility Graph](https://mlsanity.com/paper/2407.00742/) (2024)

## Abstract (arXiv)

We present a Deep Differentiable Simplex Layer (DDSL) for neural networks for geometric deep learning. The DDSL is a differentiable layer compatible with deep neural networks for bridging simplex mesh-based geometry representations (point clouds, line mesh, triangular mesh, tetrahedral mesh) with raster images (e.g., 2D/3D grids). The DDSL uses Non-Uniform Fourier Transform (NUFT) to perform differentiable, efficient, anti-aliased rasterization of simplex-based signals. We present a complete theoretical framework for the process as well as an efficient backpropagation algorithm. Compared to previous differentiable renderers and rasterizers, the DDSL generalizes to arbitrary simplex degrees and dimensions. In particular, we explore its applications to 2D shapes and illustrate two applications of this method: (1) mesh editing and optimization guided by neural network outputs, and (2) using DDSL for a differentiable rasterization loss to facilitate end-to-end training of polygon generators. We are able to validate the effectiveness of gradient-based shape optimization with the example of airfoil optimization, and using the differentiable rasterization loss to facilitate end-to-end training, we surpass state of the art for polygonal image segmentation given ground-truth bounding boxes.