  Topic modeling is a widely utilized tool in text analysis. We investigate the
optimal rate for estimating a topic model. Specifically, we consider a scenario
with $n$ documents, a vocabulary of size $p$, and document lengths at the order
$N$. When $N\geq c\cdot p$, referred to as the long-document case, the optimal
rate is established in the literature at $\sqrt{p/(Nn)}$. However, when
$N=o(p)$, referred to as the short-document case, the optimal rate remains
unknown. In this paper, we first provide new entry-wise large-deviation bounds
for the empirical singular vectors of a topic model. We then apply these bounds
to improve the error rate of a spectral algorithm, Topic-SCORE. Finally, by
comparing the improved error rate with the minimax lower bound, we conclude
that the optimal rate is still $\sqrt{p/(Nn)}$ in the short-document case.
