Two important problems in the field of Topological Data Analysis are defining practical multifiltrations on objects and showing ability of TDA to detect the geometry. Motivated by the problems, we constuct three multifiltrations named multi-GENEO, multi-DGENEO and mix-GENEO, and prove the stability of both the interleaving distance and multiparameter persistence landscape of multi-GENEO with respect to the pseudometric of the subspace of bounded functions. We also give the estimations of upper bound for multi-DGENEO and mix-GENEO. Finally, we provide experiment results on MNIST dataset to demonstrate our bifiltrations have ability to detect geometric and topological differences of digital images.

翻译：拓扑数据分析领域的两个重要问题是：如何在对象上定义实用的多重过滤方法，以及如何证明TDA具备检测几何结构的能力。受此问题驱动，我们构建了三种多重过滤方法——multi-GENEO、multi-DGENEO和mix-GENEO，并证明了multi-GENEO在交叠距离与多参数持续景观关于有界函数子空间伪度量下的稳定性。同时给出multi-DGENEO与mix-GENEO的上界估计。最后，我们在MNIST数据集上提供了实验验证，表明所提出的双参数过滤方法能够有效检测数字图像的几何与拓扑差异。