Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Weiler, Maurice; Forré, Patrick; Verlinde, Erik; Welling, Max

doi:10.48550/arxiv.2106.06020

Cited by 18 publications

(28 citation statements)

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“…has to hold. This constraint was proven in [27] for orthogonal G and later in [29] for general G. The general case also follows immediately from the distributional framework we develop in Appendix D.…”

Section: The Steerability Constraint For Kernelssupporting

confidence: 52%

Steerable Partial Differential Operators for Equivariant Neural Networks

Jenner¹,

Weiler²

2021

Preprint

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Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a G-steerability constraint that completely characterizes when a PDO between feature vector fields is equivariant, for arbitrary symmetry groups G. We then fully solve this constraint for several important groups. We use our solutions as equivariant drop-in replacements for convolutional layers and benchmark them in that role. Finally, we develop a framework for equivariant maps based on Schwartz distributions that unifies classical convolutions and differential operators and gives insight about the relation between the two. However, one remaining difference is that physics uses equivariant partial differential operators (PDOs) to define maps between fields, such as the gradient or Laplacian. Therefore, using PDOs instead of convolutions in deep learning would complete the analogy to physics and could lead to even more transfer of ideas between subjects.Equivariant PDO-based networks have already been designed in prior work [21][22][23]. Most relevant for our work are PDO-eConvs [22], which can be seen as the PDO-analogon of group convolutions. * Work done during an internship at QUVA Lab Preprint. Under review.

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Section: The Steerability Constraint For Kernelssupporting

confidence: 52%

Steerable Partial Differential Operators for Equivariant Neural Networks

Jenner¹,

Weiler²

2021

Preprint

Self Cite

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“…In this vein, it could also be interesting to derive anisotropic pooling and unpooling layers based on anisotropic spaces instead of isotropic ones as it is usually done. More generally, our method is simple enough to be extended to shapes/surfaces with a Riemannian manifold structure [Cohen et al, 2019], following the framework of coordinate independent convolutions [Weiler et al, 2021]. In this work, we focused on 2D images and spherical data on, but the method is readily extendable to higher dimensional Lie groups such as the SE(3) group to obtain 3D rototranslation equivariant ChebLieNets.…”

Section: Discussionmentioning

confidence: 99%

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

Aguettaz¹,

Bekkers²,

Defferrard³

2021

Preprint

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We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds. Surfing on the success of graph-and group-based neural networks, we take advantage of the recent developments in the geometric deep learning field to derive a new approach to exploit any anisotropies in data. Via discrete approximations of Lie groups, we develop a graph neural network made of anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global pooling layers. Group equivariance is achieved via equivariant and invariant operators on graphs with anisotropic left-invariant Riemannian distance-based affinities encoded on the edges. Thanks to its simple form, the Riemannian metric can model any anisotropies, both in the spatial and orientation domains. This control on anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic metric) against invariance (isotropic metric) of the graph convolution layers. Hence we open the doors to a better understanding of anisotropic properties. Furthermore, we empirically prove the existence of (data-dependent) sweet spots for anisotropic parameters on CIFAR10. This crucial result is evidence of the benefice we could get by exploiting anisotropic properties in data. We also evaluate the scalability of this approach on STL10 (image data) and ClimateNet (spherical data), showing its remarkable adaptability to diverse tasks.Preprint. Under review.

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“…As this note was being concluded, a helpful review and proposal for bringing more coherence to the growing body of work on geometric deep learning appeared [8], as well as a comprehensive treatment of G-equivariant CNNs [60].…”

Section: Related Workmentioning

confidence: 99%

Turing approximations, toric isometric embeddings & manifold convolutions

Suárez-Serrato

2021

Preprint

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Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we define a convolution operator for a manifold of arbitrary topology and dimension. We also explain geometric and topological conditions that make some local definitions of convolutions which rely on translating filters along geodesic paths on a manifold, computationally intractable. A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution on computable, finite metric space approximations to a smooth manifold.We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time.

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Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Cited by 18 publications

References 0 publications

Steerable Partial Differential Operators for Equivariant Neural Networks

Steerable Partial Differential Operators for Equivariant Neural Networks

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

Turing approximations, toric isometric embeddings & manifold convolutions

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