Intrinsic Mesh CNNs

Andreas Mazur, Fabian Hinder, Barbara Hammer

Primary Area: representation learning for computer vision, audio, language, and other modalities

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Keywords: Geometric Deep Learning, Convolutions, Geometry, Surfaces, Differential Geometry

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Abstract:

Rephrasing the convolution operation from Euclidean to non-Euclidean domains, such as graphs and surfaces, is of great interest in the context of geometric deep learning.
By elaborating on closing a theoretical gap between an existing framework for the parametric construction of non-Euclidean convolutions and a sound theoretical definition for intrinsic surface convolutions, motivated by differential geometry, we show that existing definitions for surface convolutions only differ in their prior assumptions about local surface information.
In the course of our efforts we found a canonical prior that allows for a theoretical definition of the class of Intrinsic Mesh CNNs, which captures the CNNs that operate on surfaces.
This class combines the practical advantages of the framework for the parametric construction of non-Euclidean convolutions with a substantiated theory, that allows for further theoretical analysis and interesting research questions.
Eventually, we conduct an experimental investigation of the canonical prior, the results of which confirm our theory about its canonical nature.

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Submission Number: 3407