Nelson-Nguyen猜想：通过均值到矩集中性

中文摘要 AI 辅助

一个 oblivious 子空间嵌入（OSE）是一个矩阵上的分布，它近似保持任意固定低维子空间中每个向量的平方欧几里得范数。我们证明了 Nelson-Nguyen 猜想：对于每个 $0 < \delta < 1$，存在一个分布，给出嵌入维数为 $O((d + \log(1/\delta))/\varepsilon^2)$、列稀疏度为 $s = O(\log(d/\delta)/\varepsilon)$ 的 OSE，失败概率至多为 $\delta$。我们首先使用迹矩论证来界定平均谱误差，然后通过集中性和重采样将该界提升到所需的高概率保证。ChatGPT-5.6-Pro 被用于证明和撰写本手稿的结果。

英文摘要

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.

发表机构

• Adobe Research（Adobe研究院）

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