Uncovering hidden relations in complex data sets is a key step to making sense of the data, which is a hot topic in our era of data deluge. Graph-based representations are examples of tools able to encode relations in a mathematical structure enabling the uncovering of patterns like clusters and paths. However, graphs only capture pairwise relations encoded in the presence of edges, but there are many forms of interaction that cannot be reduced to pairwise relations. To overcome the limitations of graph-based representations, it is necessary to incorporate multiway relations. In this article, we exploit tools from algebraic topology to handle multiway relations. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study a topological space, that is, a set of points, along with a set of neighborhoods. More specifically, we illustrate topological signal processing (TSP), a framework encompassing a class of methods for analyzing signals defined over a topological space. Given its generality, TSP incorporates graph signal processing (GSP) as a particular case. After motivating the use of topological and geometrical methods for detecting patterns in the data, we present the signal processing tools based on algebraic topology and then illustrate their advantages with respect to graph-based methodology.

中文翻译：

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拓扑信号处理：在多路关系上理解数据构建

发现复杂数据集中隐藏的关系是理解数据的关键步骤，这是我们数据泛滥时代的热门话题。基于图的表示是能够在数学结构中对关系进行编码的工具的示例，从而能够揭示集群和路径等模式。然而，图仅捕获在存在边的情况下编码的成对关系，但有许多交互形式不能简化为成对关系。为了克服基于图的表示的局限性，有必要结合多路关系。在本文中，我们利用代数拓扑中的工具来处理多路关系。代数拓扑是数学的一个分支，它使用抽象代数的工具来研究拓扑空间，即一组点以及一组邻域。更具体地说，我们说明了拓扑信号处理 (TSP)，它是一个框架，包含一类用于分析在拓扑空间上定义的信号的方法。鉴于其普遍性，TSP 将图形信号处理 (GSP) 作为一种特殊情况。在推动使用拓扑和几何方法检测数据中的模式后，我们介绍了基于代数拓扑的信号处理工具，然后说明它们相对于基于图的方法的优势。