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

• URL: http://arxiv.org/abs/2508.14234v1
• Date: Tue, 19 Aug 2025 19:45:52 GMT
• Title: Optimal Subspace Embeddings: Resolving Nelson-Nguyen Conjecture Up to Sub-Polylogarithmic Factors
• Authors: Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong,
• Abstract summary: We give a proof of the conjecture of Nelson and Nguyen [FOCS 2013] on the optimal dimension and sparsity of oblivious subspace embeddings.<n>For any $ngeq d$ and $epsilongeq d-O(1)$, there is a random $tilde O(d/epsilon2)times n$ matrix $Pi$ with $tilde O(log(d)/epsilon)$ non-zeros per column such that for any $AinmathbbRntimes d
• Score: 3.9657575162895196
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\epsilon\geq d^{-O(1)}$, there is a random $\tilde O(d/\epsilon^2)\times n$ matrix $\Pi$ with $\tilde O(\log(d)/\epsilon)$ non-zeros per column such that for any $A\in\mathbb{R}^{n\times d}$, with high probability, $(1-\epsilon)\|Ax\|\leq\|\Pi Ax\|\leq(1+\epsilon)\|Ax\|$ 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 $\tilde O(d/\epsilon^2)$ 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].

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