Speaker: Vincent Rivoirard (CEREMADE – Université Paris Dauphine)

Title: Lasso for multivariate Hawkes processes

Abstract :
Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this talk, we consider multivariate Hawkes processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive l1-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Non-asymptotic probabilistic results are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we carry out a simulation study. We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.