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Statistical learning with Hawkes processes and new matrix concentration inequalities

We consider the problem of unveiling the implicit network structure of user interactions in a social network, based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $\ell_1$ and trace-norms. We provide a first theoretical analysis of the generalization error for this problem, that includes sparsity and low-rank inducing priors. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest. A consequence of our analysis is the construction of sharply tuned penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the significant improvements achieved by the use of such data-driven penalizations on the MemeTracker dataset.

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