A Near-Optimal Lower Bound for $\ell_p$-Subspace Embeddings, $1\leq p<2$

Abstract

For $d \geq 2$, $p \geq 1$ and $\epsilon > 0$, let $N_p(d,\epsilon)$ be the smallest integer $N$ such that for every integer $n$ and every $A\in\mathbb{R}^{n\times d}$, there exists a matrix $\Phi\in\mathbb{R}^{N\times n}$ satisfying $(1-\epsilon)\lVert Ax\rVert_p\leq \lVert\Phi A x\rVert_p\leq (1+\epsilon)\lVert Ax\rVert_p$ for all $x\in\mathbb{R}^d$. For every constant $p\geq 1$ with $p\not\in 2\mathbb{Z}$, when $d\gtrsim_p \log(1/\epsilon)$, the bound \[ N_p(d,\epsilon) \gtrsim_{p} \frac{d}{\epsilon^2 \operatorname{polylog}(d/\epsilon)} \] is established. This improves the previous lower bound $\Omega(1/(\epsilon^2\operatorname{polylog}(1/\epsilon)))$ due to Li et al. (SICOMP 2021) and is optimal up to logarithmic factors for $1\leq p<2$. The central technical idea originated from ChatGPT 5.6 Sol.