Dr. Michael Perlmutter Profile

Dr. Michael Perlmutter

at Michigan State Univ

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Proceedings Article | 9 September 2019 Paper

Geometric wavelet scattering on graphs and manifolds

Feng Gao, Matthew Hirn, Michael Perlmutter, Guy Wolf

Proceedings Volume 11138, 111380Q (2019) https://doi.org/10.1117/12.2529615

KEYWORDS: Wavelets, Scattering, Wavelet transforms, Diffusion, Convolution, Signal processing, Convolutional neural networks, Data modeling, Social networks, Algorithm development, Artificial neural networks

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Convolutional neural networks (CNNs) are revolutionizing imaging science for two- and three-dimensional images over Euclidean domains. However, many data sets are intrinsically non-Euclidean and are better modeled through other mathematical structures, such as graphs or manifolds. This state of affairs has led to the development of geometric deep learning, which refers to a body of research that aims to translate the principles of CNNs to these non-Euclidean structures. In the process, various challenges have arisen, including how to define such geometric networks, how to compute and train them efficiently, and what are their mathematical properties.

In this letter we describe the geometric wavelet scattering transform, which is a type of geometric CNN for graphs and manifolds consisting of alternating multiscale geometric wavelet transforms and nonlinear activation functions. As the name suggests, the geometric wavelet scattering transform is an adaptation of the Euclidean wavelet scattering transform, first introduced by S. Mallat, to graph and manifold data. Like its Euclidean counterpart, the geometric wavelet scattering transform has several desirable properties. In the manifold setting these properties include isometric invariance up to a user specified scale and stability to small diffeomorphisms. Numerical results on manifold and graph data sets, including graph and manifold classification tasks as well as others, illustrate the practical utility of the approach.

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