Spherical Parametrized Manifold Convolutions

In geometric deep learning, manifold convolution aims to extend the convolution operator from CNNs to non-Euclidean domains. In this field, equivariance plays an important role as the target property that allows convolution filters to be applied at different locations on the manifold; for CNNs, this corresponds to translation symmetry. In curved spaces such as smooth 2D surfaces, this symmetry group is no longer valid. Therefore, the goal of equivariant convolution theory is to bridge this gap. Current approaches typically either impose restrictive topological or symmetry assumptions to obtain exact equivariance, or abandon global symmetry in favor of local geometric constructions. The former limits applicability to real-world objects with diverse shapes, while the latter must address gauge equivariance and often requires expensive differential-geometric computations. In this work, we propose to combine both worlds: equivariant convolution on general manifolds and the canonical geometry of the sphere, which offers closed-form formulas for computing connections, exponential maps, and geodesics. We use a conformal spherical parametrization, which introduces minimal distortion while allowing to precisely understand the approximation errors. The kernels are polar splines applied across rotational orientation banks, which satisfy equivariance under discrete rotations, while the pullback to the manifold accounts for the local distortion induced by the spherical parametrization. We study the behavior of our method through targeted experiments on a custom synthetic dataset, and show that the convolution is approximately equivariant under rotations. Further, our method maintains competitive performance under deformation, compared to other methods, while offering greater flexibility.