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I'm trying to figure out how $\mu$ (the coherence) behaves asymptotically for a Gaussian matrix $A\in\mathbb{R}^{n\times p}$. To generate such a normalized matrix (thanks to the concentration of norm of high-dimensional normal vectors) we can draw each element i.i.d as $N(0,\frac{1}{n})$. The inner product of such two vectors is then a sum of products of such normal RVs. Each product distributes (see here) as a difference of two $\frac{1}{n}\chi^2(1)$ independent RVs, thus the inner product distributes as the the difference of two $\frac{1}{n}\chi^2(n)$. Asymptotically $$\frac{1}{n}\chi^2(n) \rightarrow N\left(1,\frac{2}{n}\right)$$ (see here), thus their difference distributes as $$N\left(1-1,\frac{2}{n}+\frac{2}{n}\right)=N\left(0,\frac{4}{n}\right)$$

If we have $p$ columns then there are $\binom{p}{2}$ correlation coefficients that distribute normally, and they are mostly independent (the dependence is with $p-1$ other coefficients, while the number of coefficients is $~p^2$). This, along with the concentration of maximum of Gaussian ats $\sigma\sqrt{2\log{p^2}}$ gives us the expected coherence: $$4\sqrt{\frac{\log{p}}{n}}$$

However, I have good reasons to believe (both from experimental results for the coherence and this thread) that the $4$ factor in the variance is wrong, and it should be $N\left(0,\frac{1}{n}\right)$. I can't find my false derivation, though. Any hint will be much appreciated.

galra
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