🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.000967s)

  1. Differentiable Euler Characteristic Transforms for Shape Classification (2023)

    Ernst Röell, Bastian Rieck
    Abstract The _Euler Characteristic Transform_ (ECT) is a powerful invariant, 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.

  3. Simplicial Representation Learning With Neural \$K\$-Forms (2023)

    Kelly Maggs, Celia Hacker, Bastian Rieck
    Abstract Geometric deep learning extends deep learning to incorporate information about the geometry and topology data, especially in complex domains like graphs. Despite the popularity of message passing in this field, it has limitations such as the need for graph rewiring, ambiguity in interpreting data, and over-smoothing. In this paper, we take a different approach, focusing on leveraging geometric information from simplicial complexes embedded in \$\mathbb\R\\textasciicircumn\$ using node coordinates. We use differential \$k\$-forms in \$\mathbb\R\\textasciicircumn\$ to create representations of simplices, offering interpretability and geometric consistency without message passing. This approach also enables us to apply differential geometry tools and achieve universal approximation. Our method is efficient, versatile, and applicable to various input complexes, including graphs, simplicial complexes, and cell complexes. It outperforms existing message passing neural networks in harnessing information from geometrical graphs with node features serving as coordinates.