Optimal Subspace Embeddings: Resolving Nelson-Nguyen Conjecture Up to Sub-Polylogarithmic Factors

Abstract

We give a proof of the conjecture of Nelson and Nguyen [FOCS 2013] on the optimal dimension and sparsity of oblivious subspace embeddings, up to sub-polylogarithmic factors: For any $n\geq d$ and $ε\geq d^{-O(1)}$, there is a random $\tilde O(d/ε^2)\times n$ matrix $Π$ with $\tilde O(\log(d)/ε)$ non-zeros per column such that for any $A\in\mathbb{R}^{n\times d}$, with high probability, $(1-ε)\|Ax\|\leq\|ΠAx\|\leq(1+ε)\|Ax\|$ for all $x\in\mathbb{R}^d$, where $\tilde O(\cdot)$ hides only sub-polylogarithmic factors in $d$. Our result in particular implies a new fastest sub-current matrix multiplication time reduction of size $\tilde O(d/ε^2)$ for a broad class of $n\times d$ linear regression tasks. A key novelty in our analysis is a matrix concentration technique we call iterative decoupling, which we use to fine-tune the higher-order trace moment bounds attainable via existing random matrix universality tools [Brailovskaya and van Handel, GAFA 2024].