Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of sophisticated data encountered in challenging learning problems. Despite several appealing properties, integrating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. In particular, algorithms to average a family of PDs have only been considered recently and are known to be computationally prohibitive. We propose in this article a tractable framework to carry out fundamental tasks on PDs, namely evaluating distances, computing barycenters and carrying out clustering. This framework builds upon a formulation of PD metrics as optimal transport (OT) problems, for which recent computational advances, in particular entropic regularization and its convolutional formulation on regular grids, can all be leveraged to provide efficient and (GPU) scalable computations. We demonstrate the efficiency of our approach by carrying out clustering on PDs at scales never seen before in the literature.

通过将坚持图像学度量重新表述为最优传输问题，我们提出了一种可行的框架，在大规模上执行标准的坚持图像学任务，例如评估距离、估计重心和执行聚类，从而实现可扩展计算。

利用最优输运大规模计算持续图的均值和聚类