A Differential Geometric Theory of Coordinate Independents

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

Convolutional networks are shown to be equivariant w.r.t.

Those isometries that are symmetries of the corresponding structure group.

The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles.

The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of euclidean cnns, spherical cnns and cnns on general surfaces as specific instances of coordinate independent convolutions.

To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the m\"obius strip.

Authors

Maurice Weiler, Patrick Forré, Erik Verlinde, Max Welling