Let $n,w\in\mathbb{N}$, and consider the matrix ${\bf R}\in\mathbb{Z}^{2n\times w}$ sampled from a gaussian distribution with standard deviation $s>\sqrt{n}$. What I'm curious about is how small the first singular value of $\bf R$ be, i.e. how to minimize $$s_1(\bf R)=\max_{||u||=1}\{||\bf Ru||\}$$ for a certain amount of samples or given a certain value $\epsilon$ what are the chances of reaching $s_1(\bf R)=\epsilon$ or close enough. Also, how does the choice of $s$ influence this situation?
2026-03-04 19:33:41.1772652821
Singular Values from Sampled Matrix
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