Oscar Carlsson: Equivariant non-linear maps for neural networks on homogeneous spaces

Abstract.

The possible structures for possible linear equivariant maps on homogeneous spaces have been known for some time and under mild restrictions any linear map on features can be realised as an integral transformation. Making such a map equivariant requires putting constraints on the corresponding integration kernel where these, so-called steerability constraints, in the linear setting, have been fully characterised. However, any such result for the non-linear case has been lacking.
Inspired by the enormous success of inherently non-linear layers, such as the attention mechanism, and by the work done in the linear case, we constructed a framework specifying such constraints also for the non-linear case. In addition this results in a "cookbook" of ingredients with which one can construct new non-linear layers. These ingredients are used to obtain several layer architectures present in the literature.
In this presentation I will firstly provide some initial concepts and motivation followed by some necessary mathematical background (mostly intuitive notions of different bundles and the induced representation) as well as the Cartan mixing diagram which will be an important contextualising tool. Subsequently I will introduce the framework and show the constraints on the central objects required for equivariance. To conclude I will present the cookbook and its ingredients as well as how several existing layer architectures can be obtained from this.
 

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