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.

本文提出了一种计算框架，用于编码持续图的一组摩尔复合表达式，这些表达式是字典中原子图的加权Wasserstein重心; 其中，多尺度梯度下降方法可有效解决相应的最小化问题，混合了重心权重和原子图的优化，并利用了共享内存并行性。

持久图的Wasserstein字典