Sparsity-Dimension Trade-Offs for Oblivious Subspace Embeddings

An oblivious subspace embedding (OSE), characterized by parameters $m,n,d,ε,δ$, is a random matrix $Π\in \mathbb{R}^{m\times n}$ such that for any $d$-dimensional subspace $T\subseteq \mathbb{R}^n$, $\Pr_Π[\forall x\in T, (1-ε)\|x\|_2 \leq \|Πx\|_2\leq (1+ε)\|x\|_2] \geq 1-δ$. When an OSE has $s\le 1/2.001ε$ nonzero entries in each column, we show it must hold that $m = Ω\left(d^2/( ε^2s^{1+O(δ)})\right)$, which is the first lower bound with multiplicative factors of $d^2$ and $1/ε$, improving on the previous $Ω\left(d^2/s^{O(δ)}\right)$ lower bound due to Li and Liu (PODS 2022). When an OSE has $s=Ω(\log(1/ε)/ε)$ nonzero entries in each column, we show it must hold that $m = Ω\left((d/ε)^{1+1/4.001εs}/s^{O(δ)}\right)$, which is the first lower bound with multiplicative factors of $d$ and $1/ε$, improving on the previous $Ω\left(d^{1+1/(16εs+4)}\right)$ lower bound due to Nelson and Nguyen (ICALP 2014). This second result is a special case of a more general trade-off among $d,ε,s,δ$ and $m$.