This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [99], [101] of a dictionary of atom diagrams. We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams. Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

翻译：本文提出了一种计算框架，用于以加权Wasserstein重心[99][101]的形式对持续图集合进行简洁编码，其中重心由原子图词典构成。我们引入了一种多尺度梯度下降方法，通过交替优化重心权重与原子图来高效求解对应的最小化问题。该方法利用两个子问题梯度的解析表达式确保快速迭代，并进一步利用共享内存并行性加速计算。在公开数据集上的大量实验表明，该方法具有高效性——最大规模示例的Wasserstein词典计算仅需数分钟。我们通过两个应用展示了研究成果的实用性：首先，将Wasserstein词典应用于数据缩减，通过词典中简洁表示的权重可靠压缩持续图；其次，提出基于少量原子（通常三个）构建Wasserstein词典的降维框架，并将词典编码为嵌入可视化空间（通常为二维）的低维单纯形。在两个应用中，定量实验均验证了该框架的有效性。最后，我们提供可复现结果的C++实现代码。