Schatten p norm p>1

Question

The Schatten p norm is differentiable away from the origin for p> 1. Does a stronger condition of Lipschitz continuity of the gradient also hold?

Answer 1

Since any norm is homogeneous of degree 1, its gradient is homogeneous of degree 0 where it exists. Therefore this gradient cannot be Lipschitz at the origin.

However, the $p$th power of the norm certainly has an analytic gradient if $p = 2$, since that is just the Frobenius norm. In general, $A \mapsto \|A\|_p$ has a gradient that is homogeneous of degree $p-1$. This should imply that the gradient of this map is locally Lipschitz iff $p \ge 1$. It cannot be globally Lipschitz if $p > 2$.