Flexible estimation in temporal point processes and graphs

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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.