Compression of Currents and Varifolds

Abstract. We derive an algorithm for compression of the currents and varifolds representations of shapes using Ridge Leverage Score sampling and the theory of Nystrom approximation in Reproducing Kernel Hilbert Spaces. Our method is faster than existing compression techniques and comes with theoretical guarantees on the rate of convergence of the compressed approximation as a function of the smoothness of the associated shape representation. The obtained compressions are shown to be useful for down-line tasks such as nonlinear shape registration in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets and is shown to be fast and scalable with rapid error decay.