# #StackBounty: #regression #lasso #convergence #high-dimensional High-dimensional regression: why is \$log p/n\$ special?

### Bounty: 100

I am trying to read up on the research in the area of high-dimensional regression; when \$p >> n\$. It seems like the term \$log p/n\$ appears often in terms of rate of convergence for regression estimators.

For example, here, equation (17) says that the lasso fit, \$hat{beta}\$ satisfies
\$\$ dfrac{1}{n}|Xhat{beta} – X beta|_2^2 = O_P left(sigma sqrt{dfrac{log p}{n} } |beta|_1right),.\$\$

Usually, this also implies that \$log p\$ should be smaller than \$n\$.

1. Is there any intuition as to why this ratio of \$log p/n\$ is so prominent?
2. Also, it seems from the literature the high-dimensional regression problem gets complicated when \$log p geq n\$. Why is it so?
3. Is there a good reference that discusses the issues with how fast \$p\$ and \$n\$ should grow compared to each other?

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