  Stabilizing an unknown dynamical system is one of the central problems in control theory. In this paper, we study the sample complexity of the learn-to-stabilize problem in Linear Time-Invariant (LTI) systems on a single trajectory. Current state-of-the-art approaches require a sample complexity linear in $n$, the state dimension, which incurs a state norm that blows up exponentially in $n$. We propose a novel algorithm based on spectral decomposition that only needs to learn "a small part" of the dynamical matrix acting on its unstable subspace. We show that, under proper assumptions, our algorithm stabilizes an LTI system on a single trajectory with $\tilde{O}(k)$ samples, where $k$ is the instability index of the system. This represents the first sub-linear sample complexity result for the stabilization of LTI systems under the regime when $k = o(n)$. 
On the Sample Complexity of Stabilizing LTI Systems on a Single   Trajectory