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For the $n\times p$ matrix $\mathbf{X}$, is there any use in approximating $(\mathbf{X}'\mathbf{X})^{-1}$ when $p>>n$? If so, what information might this tell us?

I understand when $p, $\mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'$ is the linear projection operator on to the $p$-dimensional subspace. In the $p>>n$ case, this matrix would have to be approximated since it cannot be computed directly. However, I'm not sure how if it supplies any useful information.

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    What is p and what is n?2012-11-17
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    p is the number of parameters and n is the number of observations2012-11-17
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    $X^tX$ does not have an inverse: the rank of $X$ (and $X^t$) is at most $n$ ($), and therefore, $X^tX$ has at most rank $n$ (and therefore, isn't full rank). So, you have to be a bit more clear about what you mean by inverting this matrix, to begin with.2012-11-17

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