Modifications to a neural networks input and output layers are often required to accommodate the specificities of most practical learning tasks. However, the impact of such changes on architectures approximation capabilities is largely not understood. We present general conditions describing feature and readout maps that preserve an architectures ability to approximate any continuous functions uniformly on compacts. As an application, we show that if an architecture is capable of universal approximation, then modifying its final layer to produce binary values creates a new architecture capable of deterministically approximating any classifier. In particular, we obtain guarantees for deep CNNs and deep feed-forward networks. Our results also have consequences within the scope of geometric deep learning. Specifically, when the input and output spaces are Cartan-Hadamard manifolds, we obtain geometrically meaningful feature and readout maps satisfying our criteria. Consequently, commonly used non-Euclidean regression models between spaces of symmetric positive definite matrices are extended to universal DNNs. The same result allows us to show that the hyperbolic feed-forward networks, used for hierarchical learning, are universal. Our result is also used to show that the common practice of randomizing all but the last two layers of a DNN produces a universal family of functions with probability one. We also provide conditions on a DNNs first (resp. last) few layers connections and activation function which guarantee that these layers can have a width equal to the input (resp. output) spaces dimension while not negatively affecting the architectures approximation capabilities.
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