In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local motif, the persistence diagram of the resulting convolution describes the way the motif is distributed across the simplicial complex. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in such data. Moreover, we prove that (generically speaking) for any two labeled complexes one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability, greater flexibility for data-dependent vectorizations, and reduced computational complexity for certain data types. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.

翻译：本文考虑定义在单纯复形上的数据（如图像和标记图）的拓扑特征化方法，该方法在计算持久性之前将数据与各种滤波器进行卷积。将卷积滤波器视为局部模式，所得卷积的持久性图描述了该模式在单纯复形上的分布方式。这一流程（称为卷积持久性）扩展了拓扑学观测此类数据中模式的能力。此外，我们证明（一般而言）对于任意两个标记复形，总能找到某个滤波器使其生成不同的持久性图，因此所有可能的卷积持久性图构成的集合是一个单射不变量。这一结论通过证明卷积持久性是另一种拓扑不变量——持久同调变换——的特例而得证。卷积持久性的其他优势包括：更优的稳定性、对数据依赖性向量化的更高灵活性，以及针对特定数据类型降低计算复杂度。此外，我们通过系列实验表明，即使在采用随机滤波器且仅通过记录总持久性对所得图进行向量化的情况下，卷积仍能显著提升持久性在多种分类任务中的预测能力。