Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differentiable Euler Characteristic Transforms for Shape Classification

Ernst Roell, Bastian Rieck

TL;DR

This work develops a novel computational layer that enables learning the ECT in an end-to-end fashion, and shows that this seemingly simple statistic provides the same topological expressivity as more complex topological deep learning layers.

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.

Differentiable Euler Characteristic Transforms for Shape Classification

TL;DR

This work develops a novel computational layer that enables learning the ECT in an end-to-end fashion, and shows that this seemingly simple statistic provides the same topological expressivity as more complex topological deep learning layers.

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.
Paper Structure (17 sections, 6 equations, 4 figures, 3 tables)

This paper contains 17 sections, 6 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Related Work
  4. Mathematical Background
  5. Euler characteristic.
  6. Filtrations.
  7. Methods
  8. Differentiability.
  9. Integration into deep neural networks.
  10. Computational efficiency and implementation.
  11. Experiments
  12. Learning directions for classification
  13. Optimising Euler Characteristic Transforms
  14. Learning and visualising directions.
  15. Classifying Geometric Graphs
  16. ...and 2 more sections

Figures (4)

  • Figure 1: The standard algorithm to compute the ECC for a graph, depicted in \ref{['sfig:StandardFiltration']}, calculates the vertex filtration heights and sorts them in ascending order. One then loops over each set of predefined height values and keeps a running total of the Euler Characteristic as the number of vertices minus edges with height value less than the current height. Our approach differs in that we calculate the ECC of a graph \ref{['sfig:InputGraph']} for each vertex and edge separately\ref{['sfig:ECCs']}. The sum of the curves is computed for the edges and vertices and the total is subtracted to yield the final ECC \ref{['sfig:FinalCurve']}. The advantage is a fully parallel computation, making our formulation amenable to hardware accelerations.
  • Figure 2: Overview of the computation of the Euler Characteristic Transform. \ref{['sfig:Filter Graph']}: Given a graph and a direction, we filter it with a hyperplane (here: from left to right). The nodes and edges of the induced graph are highlighted in red, and the Euler Characteristic Curve of the graph in this direction is displayed below. By the maximum extension principle, edges are added once both target and source node are below the hyperplane. \ref{['sfig:Stack Curves']}: We compute the ECC in multiple directions. The curve in \ref{['sfig:Filter Graph']} is highlighted in red. On the vertical axis, we parametrise the direction and on the horizontal axis the height. \ref{['sfig:Final Result']}: The ECCs are stacked to form an image, where the intensity denotes the Euler Characteristic. This serves as the input for machine learning algorithms.
  • Figure 3: \ref{['sfig:ECT learn directions']}: We sample a noisy point cloud from a circle (grey). Red dots show the directions, i.e. angles, used for the ECT (left: initial, right: after training). Our method DECT spreads directions properly over the unit circle, resulting in a perfect matching of the ground truth. \ref{['sfig:ECT learn point cloud']}: DECT also permits us to optimise existing point clouds to match a target ECT in an end-to-end differentiable fashion. Using two point clouds (grey: target; red: input data), we train DECT with an MSE loss between the learned ECT and the target ECT. Starting from a randomly-initialised point cloud (left), point coordinates are optimised to match the desired shape (right). Notably, this optimisation only involves the ECT, demonstrating its capabilities as a universal shape descriptor.
  • Figure 4: Accuracy on the Letter-low dataset as a function of the number of directions.