  An oblivious subspace embedding (OSE), characterized by parameters
$m,n,d,\epsilon,\delta$, is a random matrix $\Pi\in \mathbb{R}^{m\times n}$
such that for any $d$-dimensional subspace $T\subseteq \mathbb{R}^n$,
$\Pr_\Pi[\forall x\in T, (1-\epsilon)\|x\|_2 \leq \|\Pi x\|_2\leq
(1+\epsilon)\|x\|_2] \geq 1-\delta$. For $\epsilon$ and $\delta$ at most a
small constant, we show that any OSE with one nonzero entry in each column must
satisfy that $m = \Omega(d^2/(\epsilon^2\delta))$, establishing the optimality
of the classical Count-Sketch matrix. When an OSE has $1/(9\epsilon)$ nonzero
entries in each column, we show it must hold that $m =
\Omega(\epsilon^{O(\delta)} d^2)$, improving on the previous $\Omega(\epsilon^2
d^2)$ lower bound due to Nelson and Nguyen (ICALP 2014).
