Differentiable Euler Characteristic Transforms for Shape Classification
Ernst Roell, Bastian Rieck
TL;DR
This paper introduces the Differentiable Euler Characteristic Transform (DECT), a novel end-to-end learnable layer that enhances shape and graph classification by combining topological insights with deep learning.
Contribution
The paper develops a differentiable layer for the Euler Characteristic Transform, enabling task-specific learning and maintaining topological expressivity in shape classification.
Findings
DECT achieves competitive performance with complex models.
DECT is computationally efficient and fast.
Topological expressivity is preserved in the learned transform.
Abstract
The Euler Characteristic Transform (ECT) has proven to be a powerful representation, combining geometrical and topological characteristics of shapes and graphs. However, the ECT was hitherto unable to learn task-specific representations. We overcome this issue and develop a novel computational layer that enables learning the ECT in an end-to-end fashion. Our method, the Differentiable Euler Characteristic Transform (DECT), is fast and computationally efficient, while exhibiting performance on a par with more complex models in both graph and point cloud classification tasks. Moreover, we show that this seemingly simple statistic provides the same topological expressivity as more complex topological deep learning layers.
Peer Reviews
Decision·ICLR 2024 poster
The proposed method to compute differentiable ECC is straightforward and consists of simply replacing the counting of the elements above a threshold with a sum after a softmax. Nevertheless, the method has significant potential and allows not only the use of ECCs as a graph descriptor but also to investigate of the most significant direction (it is differentiable w.r.t. the ECC direction) and to ‘invert’ the descriptor and optimize directly the input graph.
The method description is not easy to follow, and many relevant details are not clear or missing. A detailed list is provided in the question section. In particular, the architecture description is a bit confusing. In “Integration into deep NN” it is written that MLP + global pooling is used to achieve rotation permutation invariance, but the architecture is then described as a CNN over a 16x16 image. Wouldn’t this break permutation invariance? A discussion about limits is missing. For instan
- I find the core idea of this paper to be interesting. Rewriting the topological formulations using more computable components like indicator functions and sigmoids is nice. - Overall, the paper has been compiled quite well. Despite the relative inaccessibility of the core material, the writing and structure are quite good. - The choice of experiments to demonstrate the benefits of the descriptor is refreshing. I particularly enjoyed the angle of investigation in sections 5.1 and 5.2, validati
- I find it hard to truly appreciate a more stronger impact of the proposed descriptor for a wide range of applications. Despite the simplicity and comparable accuracies of Table 2, it would be nice to be more direct in explaining what features of data are simply not achievable using standard feature descriptors and how the proposed contributions alleviate it. - More significantly, I see no baseline comparison with other prior topological descriptors. For eg, how do some of the methods in: (Haj
- DECT enables learning the ECT in an end-to-end fashion, overcoming the previous inability to learn task-specific representations. - The method is highly scalable and can be integrated into deep neural networks as a layer or loss term. - DECT exhibits advantageous performance in different shape classification tasks for various modalities especially graphs.
- Although ECT is theoretically injective, it happens only when the number of directions is sufficient. For example, for point cloud classification it could be the case that the number of directions is required to be no less than the cardinality of the point set, for ECT to be injective. This restricts the expressivity, especially for the application on point clouds, and explains why the results on point cloud classification is relatively weaker than graph classification. - One key contribution
Code & Models
Videos
Taxonomy
TopicsMedical Image Segmentation Techniques