B-Spline CNNs on Lie Groups

Abstract

Group convolutional neural networks (G-CNNs) can be used to improve classical CNNs by equipping them with the geometric structure of groups. Central in the success of G-CNNs is the lifting of feature maps to higher dimensional disentangled representations, in which data characteristics are effectively learned, geometric data-augmentations are made obsolete, and predictable behavior under geometric transformations (equivariance) is guaranteed via group theory. Currently, however, the practical implementations of G-CNNs are limited to either discrete groups (that leave the grid intact) or continuous compact groups such as rotations (that enable the use of Fourier theory). In this paper we lift these limitations and propose a modular framework for the design and implementation of G-CNNs for arbitrary Lie groups. In our approach the differential structure of Lie groups is used to expand convolution kernels in a generic basis of B-splines that is defined on the Lie algebra. This leads to a flexible framework that enables localized, atrous, and deformable convolutions in G-CNNs by means of respectively localized, sparse and non-uniform B-spline expansions. The impact and potential of our approach is studied on two benchmark datasets: cancer detection in histopathology slides in which rotation equivariance plays a key role and facial landmark localization in which scale equivariance is important. In both cases, G-CNN architectures outperform their classical 2D counterparts and the added value of atrous and localized group convolutions is studied in detail.

arXiv commentAccepted for publication at ICLR 2020, code available at https://github.com/ebekkers/gsplinets

References in corpus (18)

• Group Equivariant Convolutional Networks
• Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds
• On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
• 3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data
• Rotation equivariant vector field networks
• Exploiting Cyclic Symmetry in Convolutional Neural Networks
• Spherical CNNs
• Group Equivariant Capsule Networks
• 3D G-CNNs for Pulmonary Nodule Detection
• Deep Scale-spaces: Equivariance Over Scale
• Scale equivariance in CNNs with vector fields
• A General Theory of Equivariant CNNs on Homogeneous Spaces
• Oriented Response Networks
• Equivariant Transformer Networks
• Template Matching via Densities on the Roto-Translation Group
• A rotation-equivariant convolutional neural network model of primary visual cortex
• New Approximation of a Scale Space Kernel on SE(3) and Applications in Neuroimaging
• From receptive profiles to a metric model of V1

Cited by in corpus (10)

• Discovering Symbolic Models from Deep Learning with Inductive Biases
• Sampling using $SU(N)$ gauge equivariant flows
• Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data
• PDE-based Group Equivariant Convolutional Neural Networks
• A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups
• Rotation-Equivariant Deep Learning for Diffusion MRI
• Probabilistic Numeric Convolutional Neural Networks
• 3D Solid Spherical Bispectrum CNNs for Biomedical Texture Analysis
• Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups
• GEM: Group Enhanced Model for Learning Dynamical Control Systems