Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds

Convolution has been playing a prominent role in various applications in science and engineering for many years. It is the most important operation in convolutional neural networks. There has been a recent growth of interests of research in generalizing convolutions on curved domains such as manifolds and graphs. However, existing approaches cannot preserve all the desirable properties of Euclidean convolutions, namely compactly supported filters, directionality, transferability across different manifolds. In this paper we develop a new generalization of the convolution operation, referred to as parallel transport convolution (PTC), on Riemannian manifolds and their discrete counterparts. PTC is designed based on the parallel transportation which is able to translate information along a manifold and to intrinsically preserve directionality. PTC allows for the construction of compactly supported filters and is also robust to manifold deformations. This enables us to preform wavelet-like operations and to define deep convolutional neural networks on curved domains.

[1]  Leonidas J. Guibas,et al.  Wavelets on Graphs via Deep Learning , 2013, NIPS.

[2]  Lawrence D. Jackel,et al.  Handwritten Digit Recognition with a Back-Propagation Network , 1989, NIPS.

[3]  Risi Kondor,et al.  On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups , 2018, ICML.

[4]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[5]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

[6]  Geoffrey E. Hinton,et al.  ImageNet classification with deep convolutional neural networks , 2012, Commun. ACM.

[7]  Joan Bruna,et al.  Spectral Networks and Locally Connected Networks on Graphs , 2013, ICLR.

[8]  Pierre Vandergheynst,et al.  Geometric Deep Learning: Going beyond Euclidean data , 2016, IEEE Signal Process. Mag..

[9]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

[10]  Jonathan Masci,et al.  Learning shape correspondence with anisotropic convolutional neural networks , 2016, NIPS.

[11]  Joan Bruna,et al.  Deep Convolutional Networks on Graph-Structured Data , 2015, ArXiv.

[12]  I. Chavel.  Riemannian Geometry: Subject Index , 2006 .

[13]  Yoshua Bengio,et al.  Gradient-based learning applied to document recognition , 1998, Proc. IEEE.

[14]  Pierre Vandergheynst,et al.  Vertex-Frequency Analysis on Graphs , 2013, ArXiv.

[15]  Andrew W. Fitzgibbon,et al.  Real-time human pose recognition in parts from single depth images , 2011, CVPR 2011.

[16]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[17]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[18]  Brendan J. Frey,et al.  Deep learning of the tissue-regulated splicing code , 2014, Bioinform..

[19]  Luca Maria Gambardella,et al.  Deep Neural Networks Segment Neuronal Membranes in Electron Microscopy Images , 2012, NIPS.

[20]  Pierre Vandergheynst,et al.  Geodesic Convolutional Neural Networks on Riemannian Manifolds , 2015, 2015 IEEE International Conference on Computer Vision Workshop (ICCVW).

[21]  Max Welling,et al.  Spherical CNNs , 2018, ICLR.

[22]  Tara N. Sainath,et al.  Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups , 2012, IEEE Signal Processing Magazine.

[23]  Bin Dong.  Sparse Representation on Graphs by Tight Wavelet Frames and Applications , 2014, 1411.2643.

[24]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[25]  Daniel Cremers,et al.  Dense Non-rigid Shape Correspondence Using Random Forests , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[26]  Rudrasis Chakraborty,et al.  H-CNNs: Convolutional Neural Networks for Riemannian Homogeneous Spaces , 2018, ArXiv.

[27]  Quoc V. Le,et al.  Sequence to Sequence Learning with Neural Networks , 2014, NIPS.

[28]  Michael J. Black,et al.  FAUST: Dataset and Evaluation for 3D Mesh Registration , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[29]  Pierre Vandergheynst,et al.  Wavelets on Graphs via Spectral Graph Theory , 2009, ArXiv.

[30]  Daubechies,et al.  Ten Lectures on Wavelets Volume 921 , 1992 .

[31]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[32]  Pierre Vandergheynst,et al.  ShapeNet: Convolutional Neural Networks on Non-Euclidean Manifolds , 2015, ArXiv.

[33]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Y. Wang,et al.  Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes , 2012, Comput. Graph. Forum.

[35]  Xiang Zhang,et al.  OverFeat: Integrated Recognition, Localization and Detection using Convolutional Networks , 2013, ICLR.

[36]  Simon Haykin,et al.  GradientBased Learning Applied to Document Recognition , 2001 .