ERM-Lasso classification algorithm for Multivariate Hawkes Processes paths

TL;DR

This paper introduces a new ERM-Lasso based classification method for multivariate Hawkes process paths, effectively handling high-dimensional network data with sparsity assumptions, and provides theoretical convergence guarantees supported by synthetic experiments.

Contribution

It proposes a novel two-step classification approach combining support recovery via Lasso and a subsequent $L_2$-risk minimization, with proven convergence rates.

Findings

Lasso estimator accurately recovers adjacency matrix support.

Classification method achieves optimal convergence rates.

Synthetic data experiments validate theoretical results.

Abstract

We are interested in the problem of classifying Multivariate Hawkes Processes (MHP) paths coming from several classes. MHP form a versatile family of point processes that models interactions between connected individuals within a network. In this paper, the classes are discriminated by the exogenous intensity vector and the adjacency matrix, which encodes the strength of the interactions. The observed learning data consist of labeled repeated and independent paths on a fixed time interval. Besides, we consider the high-dimensional setting, meaning the dimension of the network may be large {\it w.r.t.} the number of observations. We consequently require a sparsity assumption on the adjacency matrix. In this context, we propose a novel methodology with an initial interaction recovery step, by class, followed by a refitting step based on a suitable classification criterion. To recover the support of the adjacency matrix, a Lasso-type estimator is proposed, for which we establish rates of convergence. Then, leveraging the estimated support, we build a classification procedure based on the minimization of a $L_2$-risk. Notably, rates of convergence of our classification procedure are provided. An in-depth testing phase using synthetic data supports both theoretical results.