Convolutional Filtering on Sampled Manifolds

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 filteri...

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Bibliographic Details
Main Authors: Wang, Zhiyang, Ruiz, Luana, Ribeiro, Alejandro
Format: Conference Proceeding
Language:English
Subjects:
Convolution
Filtering
graph Laplacian
Information filters
Information processing
Laplace equations
Laplace-Beltrami operator
Manifold filter
manifold signal processing
Manifolds
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Summary: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.
ISSN:2379-190X
DOI:10.1109/ICASSP49357.2023.10097267