The Euler characteristic transform (ECT) is a signature from topological data analysis (TDA) which summarises shapes embedded in Euclidean space. Compared with other TDA methods, the ECT is fast to compute and it is a sufficient statistic for a broad class of shapes. However, small perturbations of a shape can lead to large distortions in its ECT. In this paper, we propose a new metric on compact one-dimensional shapes and prove that the ECT is stable with respect to this metric. Crucially, our result uses curvature, rather than the size of a triangulation of an underlying shape, to control stability. We further construct a computationally tractable statistical estimator of the ECT based on the theory of Gaussian processes. We use our stability result to prove that our estimator is consistent on shapes perturbed by independent ambient noise; i.e., the estimator converges to the true ECT as the sample size increases.
翻译:Euler特征变换(ECT)是一种拓扑数据分析(TDA)的标志,它总结了嵌入欧几里得空间中的形状。与其他TDA方法相比,ECT的计算速度快,并且它是一种广泛适用于形状的足够统计量。然而,形状的微小扰动可能会导致其ECT出现大幅扭曲。在本文中,我们提出了一种紧凑一维形状的新度量标准,并证明了ECT在此度量标准下是稳定的。关键是,我们的结果使用曲率而不是一个基础形状的三角形大小来控制稳定性。此外,我们基于高斯过程理论构建了一种计算上可行的ECT统计估计器。我们利用稳定性结果证明这种估计器对于受独立环境噪声扰动的形状是一致的;即,随着样本量的增加,估计器收敛于真实的ECT。