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The permutation symmetry of neurons in each layer of a deep neural networkgives rise not only to multiple equivalent global minima of the loss function,but also to first-order saddle points located on the path between the globalminima. In a network of $d-1$ hidden layers with $n_k$ neurons in layers $k =1, \ldots, d$, we construct smooth paths between equivalent global minima thatlead through a `permutation point' where the input and output weight vectors oftwo neurons in the same hidden layer $k$ collide and interchange. We show thatsuch permutation points are critical points with at least $n_{k+1}$ vanishingeigenvalues of the Hessian matrix of second derivatives indicating a localplateau of the loss function. We find that a permutation point for the exchangeof neurons $i$ and $j$ transits into a flat valley (or generally, an extendedplateau of $n_{k+1}$ flat dimensions) that enables all $n_k!$ permutations ofneurons in a given layer $k$ at the same loss value. Moreover, we introducehigh-order permutation points by exploiting the recursive structure in neuralnetwork functions, and find that the number of $K^{\text{th}}$-orderpermutation points is at least by a factor$\sum_{k=1}^{d-1}\frac{1}{2!^K}{n_k-K \choose K}$ larger than the (alreadyhuge) number of equivalent global minima. In two tasks, we illustratenumerically that some of the permutation points correspond to first-ordersaddles (`permutation saddles'): first, in a toy network with a single hiddenlayer on a function approximation task and, second, in a multilayer network onthe MNIST task. Our geometric approach yields a lower bound on the number ofcritical points generated by weight-space symmetries and provides a simpleintuitive link between previous mathematical results and numericalobservations.

Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape