  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 $k = o(n)$.
