Oskar Nordenfors

Oskar Nordenfors, Fredrik Ohlsson, Axel Flinth

We investigate the optimization of neural networks on symmetric data, and compare the strategy of constraining the architecture to be equivariant to that of using data augmentation. Our analysis reveals that that the relative geometry of the admissible and the equivariant layers, respectively, plays a key role. Under natural assumptions on the data, network, loss, and group of symmetries, we show that compatibility of the spaces of admissible layers and equivariant layers, in the sense that the corresponding orthogonal projections commute, implies that the sets of equivariant stationary points are identical for the two strategies. If the linear layers of the network also are given a unitary parametrization, the set of equivariant layers is even invariant under the gradient flow for augmented models. Our analysis however also reveals that even in the latter situation, stationary points may be unstable for augmented training although they are stable for the manifestly equivariant models.

Oskar Nordenfors, Axel Flinth

In many machine learning tasks, known symmetries can be used as an inductive bias to improve model performance. In this paper, we consider learning group equivariance through training with data augmentation. We summarize results from a previous paper of our own, and extend the results to show that equivariance of the trained model can be achieved through training on augmented data in tandem with regularization.