Learning Hyperbolic Representations of Topological Features
Learning Hyperbolic Representations of Topological Features
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发表时间：
2021-03
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通讯作者：
Panagiotis Kyriakis;Iordanis Fostiropoulos;P. Bogdan
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作者：
Panagiotis Kyriakis;Iordanis Fostiropoulos;P. Bogdan
持久图的学习特定于任务的表示是拓扑数据分析和机器学习中的重要问题。但是，当前的艺术方法的当前状态因其表现性而受到限制，因为它们专注于欧几里得表示。持续图通常包含无限持久性（即基本特征）和欧几里得空间相对于非必需特征的重要性，因为它们无法为有限点分配无限距离。为了解决这个问题，我们提出了一种学习双曲线空间上持久图的表示的方法，更具体地说是在庞康卡雷球上。通过表示无限持续性的特征无限地接近球的边界，它们与非必需特征的距离接近无穷大，从而保留了它们的相对重要性。这是在不利用极高值的可学习参数的情况下实现的，因此可以将表示形式馈入下游优化方法，并以端到端的方式有效地训练。我们在图形和图像分类任务上介绍了实验结果，并表明我们方法的性能与其他最先进方法的性能相当或超过了或超过其他状态方法的性能。
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.