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This paper explores the null space properties of neural networks. We extendthe null space definition from linear to nonlinear maps and discuss thepresence of a null space in neural networks. The null space of a given neuralnetwork can tell us the part of the input data that makes no contribution tothe final prediction so that we can use it to trick the neural network. Thisreveals an inherent weakness in neural networks that can be exploited. Oneapplication described here leads to a method of image steganography. Throughexperiments on image datasets such as MNIST, we show that we can use null spacecomponents to force the neural network to choose a selected hidden image class,even though the overall image can be made to look like a completely differentimage. We conclude by showing comparisons between what a human viewer wouldsee, and the part of the image that the neural network is actually using tomake predictions and, hence, show that what the neural network ``sees'' iscompletely different than what we would expect.

Null Space Properties of Neural Networks with Applications to Image Steganography