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In this paper, we show that feedforward and recurrent neural networks exhibitan outer product derivative structure but that convolutional neural networks donot. This structure makes it possible to use higher-order information withoutneeding approximations or infeasibly large amounts of memory, and it may alsoprovide insights into the geometry of neural network optima. The ability toeasily access these derivatives also suggests a new, geometric approach toregularization. We then discuss how this structure could be used to improvetraining methods, increase network robustness and generalizability, and informnetwork compression methods.

The Outer Product Structure of Neural Network Derivatives