We provide a theoretical framework for neural networks in terms of the representation theory of quivers, thus revealing symmetries of the parameter space of neural networks. An exploitation of these symmetries leads to a model compression algorithm for radial neural networks based on an analogue of the QR decomposition. A projected version of backpropogation on the original model matches usual backpropogation on the compressed model.1 Unless specified otherwise, the term 'neural network' exclusively refers to a multilayer perceptron (MLP).

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We develop a uniform theoretical approach towards the analysis of various neural network connectivity architectures by introducing the notion of a quiver neural network. Inspired by quiver representation theory in mathematics, this approach gives a compact way to capture elaborate data flows in complex network architectures. As an application, we use parameter space symmetries to prove a lossless model compression algorithm for quiver neural networks with certain non-pointwise activations known as rescaling activations. In the case of radial rescaling activations, we prove that training the compressed model with gradient descent is equivalent to training the original model with projected gradient descent.

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