Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks. In real scenarios topological signal data might be noisy and an important task is to process these signals by improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as "square root" of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

翻译：高阶网络能够承载拓扑信号，这些变量不仅与节点相关联，还涉及边、三角形乃至单纯复形中更高维的单纯形。此类拓扑信号可描述多种真实系统，包括洋流、神经元间的突触电流以及生物输运网络。在实际场景中，拓扑信号数据可能含有噪声，因此通过提升信噪比来处理这些信号是一项重要任务。目前，拓扑信号通常被独立处理。例如，节点信号与边信号分别处理，而缺乏能够跨维度对拓扑信号进行一致性处理的算法。本文提出狄拉克信号处理算法——一种自适应、无监督的信号处理方法，能够学习以一致方式联合滤除定义在节点、边和三角形上的拓扑信号噪声。该算法基于离散狄拉克算子构建，该算子可视为高阶霍奇拉普拉斯算子的“平方根”。我们详细讨论了狄拉克算子的性质，包括其谱特征与特征向量的手性，并利用该算子构建了狄拉克信号处理框架，可滤除定义在单纯复形节点、边及三角形上的含噪信号。在合成噪声数据与海洋漂流浮标实测噪声数据上的实验表明，该算法能够高效重建真实信号，性能优于仅基于霍奇拉普拉斯算子的算法。