学习拓扑特征的双曲表示
Learning Hyperbolic Representations of Topological Features
摘要：
持久性图的任务特定表示是拓扑数据分析和机器学习中的一个重要问题。然而，现有技术的方法在它们的表现力方面受到限制，因为它们集中在欧几里得表示上。持久性图通常包含无限持久性的特征（即，本质特征），而欧几里德空间相对于非本质特征缩小了它们的重要性，因为它们不能将无限距离分配给有限点。为了解决这个问题，我们提出了一种方法来学习双曲空间上持久性图的表示，更具体地说，在庞加莱球上。通过无限接近球的边界来表示无限持久性的特征，它们与非必要特征的距离接近无穷大，从而保持了它们的相对重要性。这是在不使用非常高的可学习参数值的情况下实现的，因此可以将表示馈送到下游优化方法中，并以端到端的方式有效地训练。我们目前的实验结果图和图像分类的任务，并表明，我们的方法的性能是等同于或超过其他国家的最先进的方法的性能。
英文摘要：
Learning task-specific representations of persistence diagrams is an important problem in topological data analysis and machine learning. However, current state of the art methods are restricted in terms of their expressivity as they are focused on Euclidean representations. Persistence diagrams often contain features of infinite persistence (i.e., essential features) and Euclidean spaces shrink their importance relative to non-essential features because they cannot assign infinite distance to finite points. To deal with this issue, we propose a method to learn representations of persistence diagrams on hyperbolic spaces, more specifically on the Poincare ball. By representing features of infinite persistence infinitesimally close to the boundary of the ball, their distance to non-essential features approaches infinity, thereby their relative importance is preserved. This is achieved without utilizing extremely high values for the learnable parameters, thus the representation can be fed into downstream optimization methods and trained efficiently in an end-to-end fashion. We present experimental results on graph and image classification tasks and show that the performance of our method is on par with or exceeds the performance of other state of the art methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
