Is anyone able to point me to a reference for this?
Let the rows of $X \in \Re^{n\times d}$ be i.i.d. uniform on the sphere of radius $\sqrt{d}$ in $\Re^d$. What is the density of the eigenvalues of $X^\top X$?
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$\begingroup$ are the rows independent? closed-form expressions for arbitrary $n,d$ are unlikely; are you interested in asymptotics for $n,d\gg 1$? $\endgroup$– Carlo BeenakkerCommented Mar 20, 2023 at 18:03
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$\begingroup$ Yes, the rows are independent. I'm after the exact value for fixed $n$ and $d$. It is known when the rows are gaussian random vectors. E.g. Corollary 3.2.19 of Robb J Muirhead. Aspects of multivariate statistical theory. Vol. 197. John Wiley & Sons, 2009. I would expect it to be known in the uniform case too, right? $\endgroup$– user27182Commented Mar 20, 2023 at 20:51
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$\begingroup$ Maybe it's possible to get somewhere using theorem 1 of these notes galton.uchicago.edu/~lalley/Courses/386/ClassicalEnsembles.pdf but in my case it seems diagonal entries don't have a density $\endgroup$– user27182Commented Mar 20, 2023 at 21:22
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$\begingroup$ in your case the martrix elements are correlated, since the squares must sum to $d$, that is an essential complication compared to the case of independent Gaussians; for $d\gg 1$ the correlations can be neglected, but not for small $d$. $\endgroup$– Carlo BeenakkerCommented Mar 20, 2023 at 21:22
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$\begingroup$ Is an asymptotic result available for this setting, if an exact one is not? $\endgroup$– user27182Commented Mar 20, 2023 at 21:53
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1 Answer
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For $d\gg 1$ the elements of $X$ have independent normal distributions (mean zero, unit variance). If you also take the limit $n\rightarrow\infty$, at fixed ratio $n/d$, the eigenvalues of $X^\top X$ have the Marchenko-Pastur density.
answered Mar 20, 2023 at 22:07