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Motivated by the vast success of deep convolutional networks, there is agreat interest in generalizing convolutions to non-Euclidean manifolds. A majorcomplication in comparison to flat spaces is that it is unclear in whichalignment a convolution kernel should be applied on a manifold. The underlyingreason for this ambiguity is that general manifolds do not come with acanonical choice of reference frames (gauge). Kernels and features thereforehave to be expressed relative to arbitrary coordinates. We argue that theparticular choice of coordinatization should not affect a network's inference-- it should be coordinate independent. A simultaneous demand for coordinateindependence and weight sharing is shown to result in a requirement on thenetwork to be equivariant under local gauge transformations (changes of localreference frames). The ambiguity of reference frames depends thereby on theG-structure of the manifold, such that the necessary level of gaugeequivariance is prescribed by the corresponding structure group G. Coordinateindependent convolutions are proven to be equivariant w.r.t. those isometriesthat are symmetries of the G-structure. The resulting theory is formulated in acoordinate free fashion in terms of fiber bundles. To exemplify the design ofcoordinate independent convolutions, we implement a convolutional network onthe M\"obius strip. The generality of our differential geometric formulation ofconvolutional networks is demonstrated by an extensive literature review whichexplains a large number of Euclidean CNNs, spherical CNNs and CNNs on generalsurfaces as specific instances of coordinate independent convolutions.

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds