  In this work we compute lower Lipschitz bounds of $\ell_p$ pooling operators
for $p=1, 2, \infty$ as well as $\ell_p$ pooling operators preceded by
half-rectification layers. These give sufficient conditions for the design of
invertible neural network layers. Numerical experiments on MNIST and image
patches confirm that pooling layers can be inverted with phase recovery
algorithms. Moreover, the regularity of the inverse pooling, controlled by the
lower Lipschitz constant, is empirically verified with a nearest neighbor
regression.
