Why $L2$ loss is strictly convex if number of samples $N$ is larger than input dimension $d$?
I think an additional assumption of linear independent inputs is necessary to have a strictly convex loss function. 
Linear independence with $N&gt;d$ gives you $Xv \neq 0 ~ \forall v \neq 0$, which implies that $X^\top X$ is positive definite, as $||Xv||_2 &gt; 0 $.