Neural networks are currently the state-of-the-art for many tasks.Invariance and same-equivariance are two fundamental properties to characterize how a model reacts to transformation: equivariance is the generalization of both. Equivariance to transformations of the inputs can be necessary properties of the network for certain tasks. Data augmentation and specially designed layers provide a way for these properties to be learned by networks. However, the mechanisms by which networks encode them is not well understood.We propose several transformational measures to quantify the invariance and same-equivariance of individual activations of a network. Analysis of these results can yield insights into the encoding and distribution of invariance in all layers of a network. The measures are simple to understand and efficient to run, and have been implemented in an open-source library. We perform experiments to validate the measures and understand their properties, showing their stability and effectiveness. Afterwards, we use the measures to characterize common network architectures in terms of these properties, using affine transformations. Our results show, for example, that the distribution of invariance across the layers of a network has well a defined structure that is dependent only on the network design and not on the training process.

Invariance and Same-Equivariance Measures for Convolutional Neural Networks

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