Encoding the Euler Characteristic Transform

2026-06-09

AI Summary

This work proposes a novel continuous encoding of the Euler Characteristic Transform (ECT) to enhance neural networks’ representational capacity for shape data. By converting Euler characteristic curves along multiple directions into vertex-level variation sequences and employing a lightweight Transformer to perform both intra-directional encoding and inter-directional aggregation, the method achieves—for the first time—a vertex-contribution-based continuous ECT representation, replacing conventional discretization strategies. Systematic evaluation across six shape classification benchmarks encompassing point clouds, graphs, meshes, and cubical complexes demonstrates that this encoding attains state-of-the-art performance on five tasks. The observed improvements are primarily attributed to the proposed encoding mechanism rather than increased model capacity.

Abstract

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 five of 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.

Problem

• Euler Characteristic Transform
• shape representation
• neural encoding
• inductive bias
• topological data analysis

Innovation

• Euler Characteristic Transform
• continuous encoding
• topological representation
• inductive bias
• equivariant architecture