Sketching with Spherical Designs for Noisy Data Fitting on Spheres

This paper proposes a sketching strategy based on spherical designs, which is applied to the classical spherical basis function approach for massive spherical data fitting. We conduct theoretical analysis and numerical verifications to demonstrate the feasibility of the proposed { sketching} strategy. From the theoretical side, we prove that sketching based on spherical designs can reduce the computational burden of the spherical basis function approach without sacrificing its approximation capability. In particular, we provide upper and lower bounds for the proposed { sketching} strategy to fit noisy data on spheres. From the experimental side, we numerically illustrate the feasibility of the sketching strategy by showing its comparable fitting performance with the spherical basis function approach. These interesting findings show that the proposed sketching strategy is capable of fitting massive and noisy data on spheres.