Higher Lower Bounds for Sparse Oblivious Subspace Embeddings

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 $T\subseteq \mathbb{R}^n$, $\Pr_\Pi[\forall x\in T, (1-\epsilon)\|x\|_2 \leq \|\Pi x\|_2\leq (1+\epsilon)\|x\|_2] \geq 1-\delta$. When an OSE has $1/(9\epsilon)$ nonzero entries in each column, we show it must hold that $m = \Omega(d^2/\epsilon^{1-O(\delta)})$, which is the first lower bound with multiplicative factors of $d^2$ and $1/\epsilon$, improving on the previous $\Omega(\epsilon^{O(\delta)}d^2)$ lower bound due to Li and Liu (PODS 2022).