Support Recovery for the Drift Coefficient of High-Dimensional Diffusions

TL;DR

This paper investigates the minimal data length needed to accurately recover the support of the drift coefficient in high-dimensional stochastic differential equations, providing bounds that guide efficient learning in complex systems.

Contribution

It establishes a fundamental lower bound on sample complexity and analyzes an $ ext{l}_1$-regularized estimator, nearly matching the lower bound for sparse matrix classes.

Findings

Lower bound on sample complexity $T$ for support recovery.

Upper bound on $T$ for $ ext{l}_1$-regularized estimator.

Nearly matching bounds for sparse matrices.

Abstract

Consider the problem of learning the drift coefficient of a $p$-dimensional stochastic differential equation from a sample path of length $T$. We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem when both $p$ and $T$ can tend to infinity. In particular, we prove a general lower bound on the sample-complexity $T$ by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a $p\times p$ matrix which describes which degrees of freedom interact under the dynamics. In this case, we analyze a $\ell_1$-regularized least squares estimator and prove an upper bound on $T$ that nearly matches the lower bound on specific classes of sparse matrices.