Lasso and probabilistic inequalities for multivariate point processes

TL;DR

This paper introduces an adaptive Lasso methodology for multivariate point processes, leveraging new probabilistic inequalities to improve coefficient selection and provide robust tuning, with applications to neuronal activity inference.

Contribution

It develops a novel adaptive Lasso approach using Bernstein inequalities for martingales, with theoretical guarantees and practical validation on multivariate Hawkes processes.

Findings

Method outperforms adaptive Lasso in simulations

Provides nonasymptotic probabilistic bounds for Hawkes processes

Robust tuning procedure applicable to neuroscience data

Abstract

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-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. Nonasymptotic probabilistic results for multivariate Hawkes processes 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 finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the adaptive Lasso procedure proposed by Zou in (J. Amer. Statist. Assoc. 101 (2006) 1418-1429). We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of (J. Amer. Statist. Assoc. 101 (2006) 1418-1429), 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.