DenseNets and a bound of the Lipschitz for the activation functions. A.1 Derivation of Lipschitz constant K for the concatenation We know that a function f is K -Lipschitz if for all points v and w the following holds: d A.2 Derivation bounded Lipschitz Concatenated ReLU We define function: R! We have four different situations that can happen. For CIFAR10, the full i-DenseNets utilize 24.9M to utilize the 25.2M of Residual Flows. For ImageNet32, i-DenseNet utilizes 47.0M parameters to utilize the 47.1M of the Residual Flow.

We introduce Invertible Dense Networks (i-DenseNets), a more parameter efficient alternative to Residual Flows. The method relies on an analysis of the Lipschitz continuity of the concatenation in DenseNets, where we enforce invertibility of the network by satisfying the Lipschitz constant. We extend this method by proposing a learnable concatenation, which not only improves the model performance but also indicates the importance of the concatenated representation. Additionally, we introduce the Concatenated LipSwish as activation function, for which we show how to enforce the Lipschitz condition and which boosts performance. The new architecture, i-DenseNet, out-performs Residual Flow and other flow-based models on density estimation evaluated in bits per dimension, where we utilize an equal parameter budget. Moreover, we show that the proposed model out-performs Residual Flows when trained as a hybrid model where the model is both a generative and a discriminative model.