# Higher order lp regularization and lp least squares like regression with p>2

Is $$l_p$$ norm with $$2 used for regularization in any practical applications? Also least squares is usually used with $$l_2$$ norm squared. But are there any applications where $$l_p$$ norm is used with $$2? Please give the references if so.

• The $\ell_\infty$ norm is sometimes used to control the largest component of a residual. For example, instead of finding a least squares solution to $Ax=b$ you could minimize $\| Ax-b\|_\infty$, if you want the largest component of the residual to be as small as possible. I haven't encountered $\ell_p$ regularization with $2 < p < \infty$ in practical applications. – littleO Jan 10 at 16:52
• Yes, you are right. I forgot to mention finiteness of p. Is $\infty >p>2$ used anywhere else? – user52705 Jan 10 at 16:58
• A weighted Holder norm with $p=3/2$ appears in the market impact model in this MOSEK paper on portfolio optimization: docs.mosek.com/whitepapers/portfolio.pdf, section 2 – Michael Grant Jan 12 at 3:03
• Thanks for the reference. But it would be nice to see $2<p<\infty$ example. – user52705 Jan 14 at 12:54