Paper: Differentiable Euler Characteristic Transforms for Shape Classification (ICLR-2024)

Reviewer comment (Novelty):
(Novelty, minor) One of the main technical contributions of this work is swapping a Dirac function to a sigmoid with a hyperparameter. It is unclear if this is a non-obvious contribution.

Author response:
The computation with indicator functions is also novel. One usually does this with a loop, which is less scalable than our method. Moreover, we add differentiability with respect to the angles and with respect to the spatial coordinates, also novel and show our method is sometimes up to an order of magnitude faster than classical GNN methods. When comparing to other topological methods the speedup is even more dramatic.  

The application of the ECT to learn directions and do point cloud approximation has not been attempted before, and we believe that this is both a novel and valuable contribution to the literature.

Reviewer comment (Theory):
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.

Author response:
This is indeed true. Guaranteed injectivity depends on the cardinality of the set and dimension see [1]. For classification tasks, only the characteristics of the data distribution are needed, hence we can do with a rather coarse approximation of the ECT and still get useful results. We believe that the point cloud results can also be partially explained by the fact that we eschew an approximation in terms of additional combinatorial complexes. Nevertheless, further research into how to get more expressive representations of the ECT in the context of machine learning are still very much needed.  We would also like to stress that this is the first application of the ECT on more complex datasets, something not attempted yet (except for MNIST and some binary classification tasks) in the literature. 

The literature usually applies a coarse dimensionality reduction such as UMAP to the ECT to get a representation for an SVM. Finding better suited classification representations is still an open question. Such a pipeline does not exploit the fact that similar directions share information, and hence organizing them in a way that allows this spatial information in the representation can be exploited by, for instance, a CNN architecture, leads to better performance. The closest analogy is flattening an image to a single vector, whereby one loses the spatial information that a CNN uses to get its performance.

Reviewer comment (Theory):
To apply the method to graph learning, it requires the graph to have spatial coordinates.

Author response:
We believe that this can be overcome: if data features on the vertices are not present, one can construct them in terms of filtrations, such as node degree or curvature information. Our method will work for these types of metrics as well. Moreover, filtrations can be learned in a task-specific fashion, thus posing no strong restriction of the scope of our work. We will clarify this in the revision.

Reviewer comment (Experiments):
More significantly, I see no baseline comparison with other prior topological descriptors. For eg, how do some of the methods in: (Hajij et al., 2023; Hensel et al., 2021, (Moor et al., 2020; Trofimov et al., 2023; Vandaele et al., 2022) compare with the proposed construction in section 5.1, 5.2 and 5.3?

Author response:
We will discuss the suggested references in more detail, but they deal with different use cases than ours, namely unsupervised representation learning (in particular the ‘Topological Autoencoders’ works and follow-ups). While we believe that the ECT has the potential to be used as a loss term in unsupervised data analysis as well, we plan on postponing such an analysis for future work and believe that additional experiments in that direction would be out of scope. Instead, we restrict ourselves to graph classification and shape analysis.

Reviewer comment (Theory):
It would be valuable to elaborate more concretely on the multi-scale aspect of the descriptor. I suspect it comes as a result of the height h, but it's hard to easily make this observation in the paper. Please confirm and elaborate.

Author response:
For a multi-scale approach, one would have to define a different type of filtration function, in particular an alpha complex would much better capture this type of information. Our method as of now captures spatial information and directional information.

Reviewer comment (Reproducibility):
In the case of point clouds, how is the graph built? Do you consider all disconnected points?

Author response:
We count the number of points above a hyperplane in a direction theta and let this hyperplane go from $-\infty$ to $\infty$. We do indeed consider all disconnected points as we scale the point cloud to fit in a unit sphere. In the case of point clouds, we do not construct a simplicial complex, we only count the points above hyperplanes in each direction.

Reviewer comment (Writing):
Eq 1 is not clear: what is k in the exponent?

Author response:
The reviewer caught a typo, it is corrected—thanks! The Euler Characteristic is computed as the alternating sum of the number of simplices in dimension n.

Reviewer comment (Writing):
Eq 5 is difficult to read, especially the definition of the indicator function 1. Also, what is \sigma_k?

Author response:
We will add some extra clarification in the manuscript. A k-dimensional simplex is denoted by $\sigma_k$ and $h_\xi(\sigma_k)$ denotes the height of the k-simplex in the direction $\xi$. This new notation is hopefully clearer.