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 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 unexpressive statistic still provides the same
topological expressivity as more complex topological deep learning layers
provide.

我们开发了一种新的计算层 DECT，它能够以端到端的方式学习 Euler Characteristic Transform，并在图形和点云分类任务中展现出与复杂模型相当的性能；此外，我们还发现这种看似非表达性的统计量仍然具有与更复杂的拓扑深度学习层一样的拓扑表达能力。

可微分的欧拉特征变换用于形状分类

Differentiable Euler Characteristic Transforms for Shape Classification

In this paper we introduce a statistic, the persistent homology transform
(PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This
statistic is a collection of persistence diagrams - multiscale topological
summaries used extensively in topological data analysis. We use the PHT to
represent shapes and execute operations such as computing distances between
shapes or classifying shapes. We prove the map from the space of simplicial
complexes in $\mathbb{R}^3$ into the space spanned by this statistic is
injective. This implies that the statistic is a sufficient statistic for
probability densities on the space of piecewise linear shapes. We also show
that a variant of this statistic, the Euler Characteristic Transform (ECT),
admits a simple exponential family formulation which is of use in providing
likelihood based inference for shapes and surfaces. We illustrate the utility
of this statistic on simulated and real data.

本文引入了一种叫做持久同调变换 (PHT) 的统计方式，用于在 $R^3$ 和 $R^2$ 中建模表面和形状，并证明了该统计映射的单射性质。同时，该文还介绍了一种名为欧拉特征变换 (ECT) 的变体，并证明了其具有简单的指数族形式，可用于形状和表面的似然推理。