TY - JOUR

T1 - PDE-based Group Equivariant Convolutional Neural Networks

AU - Smets, Bart

AU - Portegies, Jim

AU - Bekkers, Erik

AU - Duits, Remco

PY - 2020/3/9

Y1 - 2020/3/9

N2 - We present a PDE-based framework that generalizes Group equivariant
Convolutional Neural Networks (G-CNNs). In this framework, a network
layer is seen as a set of PDE-solvers where the equation's geometrically
meaningful coefficients become the layer's trainable weights.
Formulating our PDEs on homogeneous spaces allows these networks to be
designed with built-in symmetries such as rotation equivariance instead
of being restricted to just translation equivariance as in traditional
CNNs. Having all the desired symmetries included in the design obviates
the need to include them by means of costly techniques such as data
augmentation. Roto-translation equivariance for image analysis
applications is the example we will be using throughout the paper. Our
default PDE is solved by a combination of linear group convolutions and
non-linear morphological group convolutions. Just like for linear
convolution a morphological convolution is specified by a kernel and
this kernel is what is being optimized during the training process. We
demonstrate how the common CNN operations of max/min-pooling and ReLUs
arise naturally from solving a PDE and how they are subsumed by
morphological convolutions. We present a proof-of-concept experiment to
demonstrate the potential of this framework in increasing the
performance of deep learning based imaging applications.

AB - We present a PDE-based framework that generalizes Group equivariant
Convolutional Neural Networks (G-CNNs). In this framework, a network
layer is seen as a set of PDE-solvers where the equation's geometrically
meaningful coefficients become the layer's trainable weights.
Formulating our PDEs on homogeneous spaces allows these networks to be
designed with built-in symmetries such as rotation equivariance instead
of being restricted to just translation equivariance as in traditional
CNNs. Having all the desired symmetries included in the design obviates
the need to include them by means of costly techniques such as data
augmentation. Roto-translation equivariance for image analysis
applications is the example we will be using throughout the paper. Our
default PDE is solved by a combination of linear group convolutions and
non-linear morphological group convolutions. Just like for linear
convolution a morphological convolution is specified by a kernel and
this kernel is what is being optimized during the training process. We
demonstrate how the common CNN operations of max/min-pooling and ReLUs
arise naturally from solving a PDE and how they are subsumed by
morphological convolutions. We present a proof-of-concept experiment to
demonstrate the potential of this framework in increasing the
performance of deep learning based imaging applications.

KW - Computer Science - Machine Learning

KW - Computer Science - Computer Vision and Pattern Recognition

KW - Mathematics - Differential Geometry

KW - Statistics - Machine Learning

M3 - Article

VL - 2020

JO - arXiv.org, e-Print Archive, Mathematics

JF - arXiv.org, e-Print Archive, Mathematics

M1 - 2001.09046

ER -