# Encoding the Euler Characteristic Transform

arXiv: https://arxiv.org/abs/2606.10824 · HTML: https://mlsanity.com/paper/2606.10824/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Spin-Weighted Spherical CNNs](https://mlsanity.com/paper/2006.10731/) (2020)
- [3D-Rotation-Equivariant Quaternion Neural Networks](https://mlsanity.com/paper/1911.09040/) (2019)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [An Invitation to the Euler Characteristic Transform](https://mlsanity.com/paper/2310.10395/) (2023)
- [Euler Characteristic Transform Based Topological Loss for Reconstructing 3D Images from Single 2D Slices](https://mlsanity.com/paper/2303.05286/) (2023)
- [Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms](https://mlsanity.com/paper/2410.02622/) (2024)
- [Point Cloud Synthesis Using Inner Product Transforms](https://mlsanity.com/paper/2410.18987/) (2024)
- [SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spheres](https://mlsanity.com/paper/2503.09456/) (2025)
- [LEAP: Local ECT-Based Learnable Positional Encodings for Graphs](https://mlsanity.com/paper/2510.00757/) (2025)
- [Molecular Machine Learning Using Euler Characteristic Transforms](https://mlsanity.com/paper/2507.03474/) (2025)
- [RotaTouille: Rotation Equivariant Deep Learning for Contours](https://mlsanity.com/paper/2508.16359/) (2025)

## Abstract (arXiv)

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on all six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.