40 On Learning Sets of Symmetric Elements

40 On Learning Sets of Symmetric Elements
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Authors: Haggai Maron, Or Litany, Gal Chechik, Ethan Fetaya
Summary: Learning from unordered sets is a fundamental learning setup, recently attracting
increasing attention. Research in this area has focused on the case where elements of the set
are represented by feature vectors, and far less emphasis has been given to the common case
where set elements themselves adhere to their own symmetries. That case is relevant to numerous
applications, from deblurring image bursts to multi-view 3D shape recognition and reconstruction.
In this paper, we present a principled approach to learning sets of general symmetric elements.
We first characterize the space of linear layers that are equivariant both to element reordering and
to the inherent symmetries of elements, like translation in the case of images. We further show
that networks that are composed of these layers, called Deep Sets for Symmetric elements layers
(DSS), are universal approximators of both invariant and equivariant functions. DSS layers are
also straightforward to implement. Finally, we show that they improve over existing set-learning
architectures in a series of experiments with images, graphs and pointclouds.