Inspired by recent work of Kondor  and Cohen and Welling , we build rotation equivariant autoencoders to obtain a basis of images adapted to the group of planar rotations SO(2), directly from the data. We do this in an unsupervised fashion, working in the Fourier domain of SO(2). Working in the Fourier domain we build a rotation equivariant classifier to classify images. As in the recent papers of Thomas et al.  and Kondor et al.  we use tensor product nonlinearity to build our autoencoders and classifiers. We discover the basis using a small sample of inputs. As a consequence our classifier is robust to rotations - the classifier trained on upright images, classifies rotated versions of images, achieving state of the art. In order to deal with images under different scales simultaneously, we define the notion of a coupled-bases and show that a coupled-bases can be learned using tensor nonlinearity.
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