# pr.chance – High-probability upper-bound for \$|(G^high G)^{-1} (1,ldots,1)|_2\$, the place \$G\$ is gaussian random matrix with iid entries from \$mathcal N(0,1/N)\$ Answer

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## pr.chance – High-probability upper-bound for \$|(G^high G)^{-1} (1,ldots,1)|_2\$, the place \$G\$ is gaussian random matrix with iid entries from \$mathcal N(0,1/N)\$

Let $$G$$ breathe an $$N occasions n$$ random matrix with iid entries from $$mathcal N(0,1/N)$$, with $$n/N =: lambda in (0, 1)$$, and let $$u=(1,1,ldots,1) in mathbb R^n$$.

Question. What is a upper-bound for $$|(G^high G)^{-1} u|$$ which holds with high-probability ?

Note {that a} trifling certain which holds w.p $$1-e^{-Omega(N)}$$ is given by
$$|(G^high G)^{-1} u| le lambda_{max}((G^high G)^{-1})|u| = sqrt{n}/lambda_{min}(G^high G) = mathcal O(sqrt{n}), tag{1}$$
since $$lambda_{min}(G^high G) = Omega(1)$$ w.p $$1-e^{-Omega(N)}$$.

Observation. Empirically, i’ve seen that the certain (1) is nice when $$lambda$$ is “small” and really lax when $$lambda$$ is “large”.

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