Problem
Research questions and friction points this paper is trying to address.
Nelson-Nguyen Conjecture
oblivious subspace embedding (OSE)
embedding dimension
column sparsity
failure probability
The Nelson-Nguyen Conjecture via Mean-to-Moments Concentration
🤖 AI Summary
本文通过均值-矩集中方法证明了Nelson-Nguyen猜想，解决了构造具有特定维度和稀疏性的盲目子空间嵌入问题。
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Nelson-Nguyen Conjecture
Oblivious Subspace Embedding (OSE)
Trace-Moment Argument
Concentration Inequality
Resampling
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