$p$ has at least $(m_{\ell-1}+1 - n) m_\ell$ zero eigenvalues.

We also show that local minima of $L$ cannot be isolated in the set of critical points, meaning that if $p$ is a local minimum, any $\epsilon$ neighborhood of $p$ must contain another local minimum.

[**Proposition \[noisolatedmin\].**]{} For a fully connected feedforward neural network with width $m$ larger than the number of training samples $n$, continuous activation function $\sigma$ and L2 loss function $L$, $L$ has no isolated local minima.

We show an analogous result for local maxima, but are unable to prove an analogous result for any other critical points, and it remains an open question whether with very wide networks $L$ can have isolated saddle points.

Finally, we compare the growth rates of the dimension of the loci of all the types of critical points we discuss in this paper, as well as the growth rates of the zero eigenspaces of the Hessian of $L$ at each type.

[**Proposition \[dimloci\].**]{} Consider a family of feedforward neural networks with hidden layers of increasing width training on a fixed data set. That is, let $a, b,$ and $n$ be fixed while $m$ increases. Then the dimensions of the locus of 