Flexible estimation in temporal point processes and graphs

具有历史依赖性的多变量点过程（如Hawkes过程）自然地允许用图表示。节点是过程的维度，边缘出现在具有非空交互功能的维度之间。在非负情况下-考虑激励现象-这些函数的非参数估计已经在Donnet等人的贝叶斯框架中进行了研究。在这种情况下，图的估计和非正情况-抑制现象的会计-是开放的话题。事实上，这些问题在神经科学领域引起了很多关注。神经元通过动作电位相互作用，这些动作电位通常被视为相同的事件（尖峰序列）。因此，多元霍克斯过程可以模拟神经元之间的功能连接，并识别兴奋/抑制关系。为了分析多变量Hawkes过程得到的图，可以通过将交互函数的范数关联到每个加权边来获得加权边。在非线性Hawkes过程的情况下，这些函数可以是非正的，并导致负加权边。生成的图被称为有符号的图，并且需要特殊的图分析方法。特别是，聚类是一种流行的任务，旨在识别网络中具有相似特征的节点社区。对于有符号的情况，谱算法已经适应，但仍然被认为是次优的。Cucuringu等人开发了一种基于正则图拉普拉斯矩阵组合的新方法。然而，关于随机图模型的理论结果只在两个不相交群体的情况下得到。扩展到更大数量的社区和稀疏网络的背景仍然是未解决的问题苏菲·唐纳，文森特·里沃拉德，还有朱迪斯·卢梭。多元Hawkes过程的非参数贝叶斯估计。arXiv: 1802.05975 v2, 2018。[2]Mihai Cucuringu， Peter Davies和Hemant Tyagi。聚类签名网络的广义特征问题。AISTATS, 2018年。

Multivariate point processes with history dependence such as Hawkes processes naturally admit a graph representation. Nodes are the dimensions of the process and edges appear between dimensions having a non-null interaction function. In the non-negative case - accounting for excitation phenomena - nonparametric estimation of those functions have been studied in the Bayesian framework in Donnet et al. [1]. In this context, estimation of the graph and the non-positive case - accounting for inhibition phenomena - are open topics. In fact, these problems encounter a lot of interest in the neuroscience context. Neurons interact through action potentials which are generally treated as identical events (spike trains). Multivariate Hawkes Processes can thus model functional connectivity between neurons and identify excitation/inhibition relationships. To analyse the graph obtained from Multivariate Hawkes processes, weighted edges can be obtained by associating to each of them a norm of the interaction functions. In the case of non-linear Hawkes processes, those functions are allowed to be non-positive and lead to negatively weighted edges. The resulting graph is said to be signed, and requires special methods of graph analysis. In particular, clustering is a popular task which aims at identifying communities of nodes having similar features within a network. For the signed case, spectral algorithms have been adapted but are still considered suboptimal. Cucuringu et al. [2] developed a new method based on the combination of regularized graph Laplacian matrices. However, theoretical results on random graph models have only been obtained in the case of two disjoint communities. Extension to a larger number of communities and the context of sparse networks are still unanswered questions.[1] Sophie Donnet, Vincent Rivoirard, and Judith Rousseau. Nonparametric Bayesian Estimation of Multivariate Hawkes Processes. arXiv:1802.05975v2, 2018.[2] Mihai Cucuringu, Peter Davies and Hemant Tyagi. SPONGE: A generalized eigenproblem for clustering signed networks. AISTATS, 2018.