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 $s\le 1/2.001\epsilon$ nonzero entries in each column, we show it must hold that $m = \Omega\left(d^2/( \epsilon^2s^{1+O(\delta)})\right)$, which is the first lower bound with multiplicative factors of $d^2$ and $1/\epsilon$, improving on the previous $\Omega\left(d^2/s^{O(\delta)}\right)$ lower bound due to Li and Liu (PODS 2022). When an OSE has $s=\Omega(\log(1/\epsilon)/\epsilon)$ nonzero entries in each column, we show it must hold that $m = \Omega\left((d/\epsilon)^{1+1/4.001\epsilon s}/s^{O(\delta)}\right)$, which is the first lower bound with multiplicative factors of $d$ and $1/\epsilon$, improving on the previous $\Omega\left(d^{1+1/(16\epsilon s+4)}\right)$ lower bound due to Nelson and Nguyen (ICALP 2014). This second result is a special case of a more general trade-off among $d,\epsilon,s,\delta$ and $m$.

翻译：遗忘子空间嵌入（OSE），由参数 $m,n,d,\epsilon,\delta$ 表征，是一个随机矩阵 $\Pi\in \mathbb{R}^{m\times n}$，满足：对于任意 $d$ 维子空间 $T\subseteq \mathbb{R}^n$，有 $\Pr_\Pi[\forall x\in T, (1-\epsilon)\|x\|_2 \leq \|\Pi x\|_2\leq (1+\epsilon)\|x\|_2] \geq 1-\delta$。当OSE每列的非零元个数 $s\le 1/2.001\epsilon$ 时，我们证明必然有 $m = \Omega\left(d^2/( \epsilon^2s^{1+O(\delta)})\right)$，这是首个同时包含 $d^2$ 和 $1/\epsilon$ 乘积因子的下界，改进了Li与Liu（PODS 2022）之前得到的 $\Omega\left(d^2/s^{O(\delta)}\right)$ 下界。当OSE每列的非零元个数 $s=\Omega(\log(1/\epsilon)/\epsilon)$ 时，我们证明必然有 $m = \Omega\left((d/\epsilon)^{1+1/4.001\epsilon s}/s^{O(\delta)}\right)$，这是首个同时包含 $d$ 和 $1/\epsilon$ 乘积因子的下界，改进了Nelson与Nguyen（ICALP 2014）之前得到的 $\Omega\left(d^{1+1/(16\epsilon s+4)}\right)$ 下界。第二个结果是 $d,\epsilon,s,\delta$ 与 $m$ 间更一般权衡关系的特例。