Flexible estimation in temporal point processes and graphs

具有历史相关性的多元点过程，如Hawkes过程，自然允许用图表示。节点是过程的维度，边缘出现在具有非零交互功能的维度之间。在非负情形下，Donnet等人在贝叶斯框架下研究了这些函数的非参数估计。[1]。在这种情况下，对曲线图和非正性案例的估计--解释抑制现象--是开放的话题。事实上，在神经科学的背景下，这些问题引起了很多人的兴趣。神经元通过动作电位相互作用，动作电位通常被视为相同的事件(棘波序列)。因此，多变量Hawkes过程可以对神经元之间的功能连接进行建模，并识别兴奋/抑制关系。为了分析从多变量Hawkes过程得到的图，可以通过将相互作用函数的范数关联到每条边来获得加权边。在非线性霍克斯过程的情况下，这些函数被允许为非正的，并导致负加权的边。所得到的图形被认为是经过签名的，并且需要特殊的图形分析方法。特别是，聚类是一项流行的任务，其目的是识别网络中具有相似特征的节点社区。对于带符号的情况，谱算法已经被采用，但仍然被认为是次优的。Cucuringu等人。[2]提出了一种基于正则化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仅在两个不相交的群落的情况下得到。扩展到更多的社区和稀疏网络的背景仍然是悬而未决的问题。[1]索菲·多内特、文森特·里沃拉德和朱迪思·卢梭。多元Hawkes过程的非参数贝叶斯估计Arxiv：1802.05975v2,2018.[2]Mihai Cucuringu，Peter Davies和Hemant Tyagi。海绵：一种用于对有符号网络进行聚类的广义特征问题。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.