基于膨胀不变瓶颈比较度量的拓扑信息检索
Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures
AI总结：
本文提出两种膨胀不变比较度量，用于在持续同调中刻画能保持数据库拓扑的嵌入，并基于此实现首个拓扑信息检索方法，在文本、视频和医学图像数据上优于标准瓶颈距离。
摘要：
在信息检索中，恰当地表示数据库中的元素以便查询能够被准确匹配是一项核心任务；近年来，这一目标已通过使用多种度量将数据库的图结构以保持层次的方式嵌入到流形中而实现。持续同调是拓扑数据分析中常用的工具，能够从层次结构和连通性结构两方面严格刻画数据库。在各种嵌入数据集上计算持续同调表明，一些常用的嵌入未能保持连通性。我们证明，那些成功保留数据库拓扑的嵌入在持续同调上是一致的，为此引入两种膨胀不变比较度量来捕捉这一效应：特别是，它们解决了流形上的度量失真问题。我们提供了一种计算这些度量的算法，其时间复杂度相比现有方法大幅降低。我们使用这些度量执行了首个基于拓扑的信息检索实例，并证明其性能优于持续同调的标准瓶颈距离。我们在包括文本、视频和医学图像在内的不同数据类型的数据库上展示了我们的方法。
英文摘要：
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in a hierarchy-preserving manner using a variety of metrics. Persistent homology is a tool commonly used in topological data analysis that is able to rigorously characterize a database in terms of both its hierarchy and connectivity structure. Computing persistent homology on a variety of embedded datasets reveals that some commonly used embeddings fail to preserve the connectivity. We show that those embeddings which successfully retain the database topology coincide in persistent homology by introducing two dilation-invariant comparative measures to capture this effect: in particular, they address the issue of metric distortion on manifolds. We provide an algorithm for their computation that exhibits greatly reduced time complexity over existing methods. We use these measures to perform the first instance of topology-based information retrieval and demonstrate its increased performance over the standard bottleneck distance for persistent homology. We showcase our approach on databases of different data varieties including text, videos, and medical images.
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。
