  We study the problem of learning to stabilize unknown noisy Linear
Time-Invariant (LTI) systems on a single trajectory. It is well known in the
literature that the learn-to-stabilize problem suffers from exponential blow-up
in which the state norm blows up in the order of $\Theta(2^n)$ where $n$ is the
state space dimension. This blow-up is due to the open-loop instability when
exploring the $n$-dimensional state space. To address this issue, we develop a
novel algorithm that decouples the unstable subspace of the LTI system from the
stable subspace, based on which the algorithm only explores and stabilizes the
unstable subspace, the dimension of which can be much smaller than $n$. With a
new singular-value-decomposition(SVD)-based analytical framework, we prove that
the system is stabilized before the state norm reaches $2^{O(k \log n)}$, where
$k$ is the dimension of the unstable subspace. Critically, this bound avoids
exponential blow-up in state dimension in the order of $\Theta(2^n)$ as in the
previous works, and to the best of our knowledge, this is the first paper to
avoid exponential blow-up in dimension for stabilizing LTI systems with noise.
