Convolutional Filtering on Sampled Manifolds

TL;DR

This paper establishes a theoretical error bound for approximating continuous manifold convolutions using sampled data and demonstrates the convergence of graph-based filters to true manifold filters through empirical validation.

Contribution

It provides the first non-asymptotic error bound for sampled manifold convolutional filters, bridging continuous and discrete manifold processing.

Findings

Convolutional filtering on sampled manifolds converges to continuous filtering.

Derived a non-asymptotic error bound for manifold convolution approximation.

Empirical validation on navigation control tasks supports theoretical results.

Abstract

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.