B-Spline CNNs on Lie groups
Group convolutional neural networks 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 is guaranteed via group theory.Currently, however, the practical implementations of G-CNNs are limited to either discrete groups or continuous compact groups such as rotations.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.
The paper describes a flexible framework for building CNNs that are equivariant to a large class of transformations groups.A framework for building group CNN with an arbitrary Lie group G, which shows superiority over a CNN in tumor classification and landmark localization. 