Convolutional Filtering on Sampled Manifolds
This work addresses the challenge of effective linear information processing on manifolds for applications like geometric data analysis, though it appears incremental as it builds on existing manifold and graph convolution frameworks.
The paper tackles the problem of approximating continuous manifold convolutional filters with graph convolutions on sampled manifolds, deriving a non-asymptotic error bound that shows convergence to continuous filtering and demonstrating it empirically on a navigation control task.
The increasing availability of geometric data has motivated the need for information processing over non-Euclidean domains modeled as manifolds. The building block for information processing architectures with desirable theoretical properties such as invariance and stability is convolutional filtering. Manifold convolutional filters are defined from the manifold diffusion sequence, constructed by successive applications of the Laplace-Beltrami operator to manifold signals. However, the continuous manifold model can only be accessed by sampling discrete points and building an approximate graph model from the sampled manifold. Effective linear information processing on the manifold requires quantifying the error incurred when approximating manifold convolutions with graph convolutions. In this paper, we derive a non-asymptotic error bound for this approximation, showing that convolutional filtering on the sampled manifold converges to continuous manifold filtering. Our findings are further demonstrated empirically on a problem of navigation control.