Signal Recovery from Pooling Representations

In this work we compute lower Lipschitz bounds of ${\mathrm{\ell }}_p$ pooling operators for $p=1,2,\mathrm{\infty }$ as well as ${\mathrm{\ell }}_p$ 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.