Encoding the Euler Characteristic Transform

Technical Summary: Encoding the Euler Characteristic Transform

Problem Statement
The Euler Characteristic Transform (ECT) is an injective shape descriptor that records the Euler characteristic of a linearly embedded cell complex as a function of filtration height across multiple directions. While the Differentiable ECT (DECT) has enabled end-to-end learning, existing neural network pipelines typically encode the ECT by discretizing the Euler Characteristic Curve (ECC) for each direction onto a fixed grid of height thresholds. This discretization introduces a resolution hyperparameter that requires tuning, trades off expressivity against computational cost, and forces uniform resolution regardless of where topological changes occur. The authors argue that this grid-based approach is suboptimal because the information in the ECC is entirely attributable to the vertices of the complex. The paper addresses the question of how the global ECT should be encoded for neural networks, proposing to replace discretization with a continuous, token-based representation.

Methodology
The authors introduce a continuous ECC encoding that eliminates the height-resolution hyperparameter. For a cell complex embedded in $\mathbb{R}^2$ and a given direction $w_i$, the filtration height of a cell $\tau$ is determined by the vertex with the maximum projection onto $w_i$, termed the "responsible vertex." The method computes the net Euler characteristic change, $\Delta\chi(v, w_i)$, attributed to each vertex $v$. This change combines the vertex’s own contribution with the signed contributions of higher-dimensional cells entering the filtration at the same threshold.

The continuous representation consists of a sequence of tokens for each direction, where each token is a 2-dimensional vector $(\langle v, w_i \rangle, \Delta\chi(v, w_i))$. Vertices with zero net change across all directions are discarded. A shared transformer encoder processes these variable-length token sequences within each direction, mapping them to a fixed-length feature vector via max-pooling. This results in a $D \times d_{out}$ matrix (where $D$ is the number of directions) that serves as the input to downstream representation architectures, replacing the traditional discretized ECT matrix.

The authors evaluate six distinct ECT representation architectures to process this encoded matrix:

1. Feedforward Network: A baseline that flattens the ECT matrix, making no structural assumptions.
2. Deep Set: Treats the ECT as an unordered set of ECCs, exploiting permutation invariance.
3. 2D Convolution: Treats the ECT matrix as a 2D image, using circular padding along the directional dimension to preserve rotational equivariance.
4. 1D Convolution: Applies 1D convolutions along the directional dimension with circular padding.
5. 1D Complex Convolution: Encodes angular information in the phase of complex numbers, using complex-valued convolutions that are inherently rotationally equivariant.
6. Hybrid: Combines 1D convolutions with circular padding followed by fully connected layers.

Key Contributions

1. Continuous Representation: Introduction of a grid-free, exact, and parsimonious continuous encoding of the ECT that records net Euler characteristic changes per vertex per direction. This representation is inherently differentiable and removes the need for resolution tuning.
2. Systematic Architecture Study: Evaluation of six neural network architectures with varying inductive biases, ranging from structure-agnostic feedforward networks to models preserving equivariance under planar rotations (convolutional and complex-valued models).
3. Empirical Evaluation: Comprehensive benchmarking across six classification tasks involving point clouds, graphs, cubical complexes, and meshes, comparing continuous and discrete encodings with respect to accuracy and training stability.

Results
Experiments were conducted on four data types: point clouds and cubical complexes derived from Fashion-MNIST, graph datasets (Letter-low, Letter-med, Letter-high) from TUDataset, and the SwissBuildings 3D mesh dataset.

• Accuracy: The continuous encoding improved accuracy on five of the six datasets. The largest gains were observed on Letter-high (+0.120) and SwissBuildings (+0.094). The only exception was Letter-low, where the discrete encoding was already near saturation.
• Source of Gain: Control experiments showed that adding a transformer to the discrete encoding did not yield consistent gains. Therefore, the performance improvement is attributed to the tokenization itself rather than the added capacity of the transformer.
• Architecture Performance: The choice of representation architecture mattered less than the encoding method. Under continuous encoding, the feedforward network was the strongest performer on five of the six datasets. Under discrete encoding, the feedforward network was unstable, whereas convolutional architectures (particularly the hybrid 1D convolution) were more robust.
• Sample Efficiency: Continuous encoding demonstrated better sample efficiency, particularly for feedforward and Deep Set representations, outperforming discrete variants across all training set sizes.

Significance and Claims
The paper claims that the continuous encoding provides a more learnable input for neural networks, allowing simpler architectures like feedforward networks to perform competitively with or better than specialized equivariant models. The authors emphasize that the payoff from inductive biases (such as rotational equivariance) depends heavily on the encoding; while convolutional architectures offer stability under discretization, the continuous encoding reduces the necessity for such structural biases in many cases. The work establishes that the encoding strategy is a critical inductive bias in ECT-based pipelines, and that a continuous, vertex-attributed representation is superior to conventional grid-based discretization for shape classification tasks. The authors note that extending this framework to 3D shapes requiring $SO(3)$-equivariance remains a non-trivial future direction.