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arXiv · 2609.22548

The Nelson-Nguyen Conjecture via Mean-to-Moments Concentration

Tung Mai·Anup Rao
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Abstract

An oblivious subspace embedding (OSE) is a distribution over matrices that approximately preserves the squared Euclidean norm of every vector in any fixed low-dimensional subspace. We prove the Nelson-Nguyen conjecture: for every $0 < δ< 1$, there exists a distribution that gives an OSE with embedding dimension $O((d + \log(1/δ))/\varepsilon^2)$ and column sparsity $s = O(\log(d/δ)/\varepsilon)$, with failure probability at most $δ$. We first bound the mean spectral error using a trace-moment argument and then upgrade this bound to the desired high-probability guarantee using concentration and resampling. ChatGPT-5.6-Pro was used in proving and writing the results of this manuscript.

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Tung Mai, Anup Rao. 2026-09-18. The Nelson-Nguyen Conjecture via Mean-to-Moments Concentration. https://arxiv.org/abs/2609.22548

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