Learning Networks of Stochastic Differential Equations

TL;DR

This paper investigates the problem of learning the structure of networks representing stochastic differential equations from observed trajectories, providing performance guarantees for an $ ext{l}_1$-regularized approach in sparse settings.

Contribution

It introduces theoretical guarantees for network inference from stochastic differential equations using $ ext{l}_1$ regularization, valid at high sampling rates.

Findings

Performance guarantees are uniform in sampling rate for sparse networks.

The results establish a notion of 'time complexity' for network inference.

The analysis applies to linear stochastic dynamics models.

Abstract

We consider linear models for stochastic dynamics. To any such model can be associated a network (namely a directed graph) describing which degrees of freedom interact under the dynamics. We tackle the problem of learning such a network from observation of the system trajectory over a time interval $T$. We analyze the $\ell_1$-regularized least squares algorithm and, in the setting in which the underlying network is sparse, we prove performance guarantees that are \emph{uniform in the sampling rate} as long as this is sufficiently high. This result substantiates the notion of a well defined `time complexity' for the network inference problem.