PDE-Based Group Equivariant Convolutional Neural Networks

Abstract

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 geometrically meaningful PDE-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 in addition to the standard translation equivariance of 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. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and nonlinear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers; we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch. Just like for linear convolution, a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs, we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning-based imaging applications with far fewer parameters than traditional CNNs.

Analysis of (sub-)Riemannian PDE-G-CNNs

“…Furthermore, we show that locally both kernels are similar through an upper bound on the relative error. This improves upon results in [28,Lem.20]. • Table 2 demonstrates qualitatively that ρ b becomes a poor approximation when the spatial anisotropy is high ζ ≫ 1.…”

“…In Theorem 1 we use this to improve upon local and global approximations of the relative errors of the erosion and dilations kernels used in the papers [28,60] where PDE-G-CNN are first proposed (and shown to outperform G-CNNs). Our new sharper estimates for distance on M 2 have bounds that explicitly depend on the metric tensor field coefficients.…”

“…In [28] partial differential equation (PDE) based G-CNNs are presented, aptly called PDE-G-CNNs. In fact, G-CNNs are shown to be a special case of PDE-G-CNNs (if one restricts the PDE-G-CNNs only to convection, using many transport vectors [28,Sec.6]). With PDE-G-CNNs the usual non-linearities that are present in current networks, such as the ReLU activation function and max-pooling, are replaced by solvers for specifically chosen non-linear evolution PDEs.…”

“…2: Overview of a PDE-G-CNN layer. The linear parts are solved by linear group convolutions with a certain kernel [28], and the non-linear parts are solved by morphological convolutions (3) with a morphological kernel (1). quantity in the analysis of this paper is the spatial anisotropy ζ := w1 w2 , as will become clear later.…”

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

“…For future work, it would be interesting to look into the possibilities to train the cost function 𝐶 using PDE-G-CNNs [69], which is now geometrically computed as explained in Appendix D. In the past, this method had promising results for vessel segmentation. Besides using PDE-G-CNNs to construct the cost function, it would be worth looking into the possibilities to use neural networks to calculate the distance function as was done in [70].…”