Need help with notation - Supremum

Question

I'm reading a paper, 'Invariant Scattering Convolution Networks'. Equation 2, page 2 reads as follows;

I'm not familiar with the $\underset{u}{sup}$ notation - does this refer to the support of the function $|\nabla{ \tau (u) }|$? How would I read that part of the equation out loud?

Answer 1

"Sup" stands for "supremum", not "support" ("support" would usually be abbreviated "supp" instead). So $$\sup_u |\nabla\tau(u)|$$ means the supremum of all values of $|\nabla\tau(u)|$; that is, the smallest number $s$ such that $|\nabla\tau(u)|\leq s$ for all $u$.

The symbol "$\sup\limits_u$" can be read aloud as "(the) sup (over $u$) of" or "the supremum (over $u$) of", where the parts in parentheses are optional and "sup" is pronounced like "soup".