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Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in expectation or (preferably) with high probability):

$$\|\Sigma^{-1}_m-\Sigma^{-1}\| \le \varepsilon \|\Sigma^{-1}\|$$

For the sample covariance it is known that $$\|\Sigma_m-\Sigma\| \le \varepsilon \|\Sigma\|$$ can be achieved with $r=\operatorname{tr} \Sigma/\|\Sigma\|$ (the intrinsic dimension) $$m \approx \varepsilon^{-2} r$$

I wonder if there is anything similar for the inverse covariance?

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  • $\begingroup$ $\Sigma_m^{-1}-\Sigma^{-1}=\Sigma^{-1}(\Sigma-\Sigma_m)\Sigma_m^{-1}$ gives a rough bound that may be sufficient for some problems. $\endgroup$
    – jlewk
    Commented Feb 7 at 2:17

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