Symmetric Linear Dynamical Systems are Learnable from Few Observations

TL;DR

This paper introduces a new method for learning symmetric linear dynamical systems from very few observations, achieving accurate recovery with only logarithmic sample complexity, applicable to both sparse and dense matrices.

Contribution

The authors propose a novel estimator based on the method of moments that accurately recovers symmetric matrices from a single trajectory with minimal observations, without regularization.

Findings

Achieves small element-wise error in matrix recovery

Requires only O(log N) observations regardless of matrix sparsity

Applicable to structure discovery in dynamical systems

Abstract

We consider the problem of learning the parameters of a $N$-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time $T$. We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only $T=\mathcal{O}(\log N)$ observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.