Erik Bekkers

Erik Bekkers is an assistant professor in Geometric Deep Learning in the Machine Learning Lab of the University of Amsterdam (AMLab, UvA). Before this he did a post-doc in applied differential geometry at the dept. of Applied Mathematics at Technical University Eindhoven (TU/e). In his PhD thesis (cum laude, Biomedical Engineering, TU/e), he developed medical image analysis algorithms based on sub-Riemannian geometry in the Lie group SE(2) using the same mathematical principles that underlie mathematical models of human visual perception. Such mathematics find their application in machine learning where through symmetries and geometric structure, robust and efficient representation learning methods are obtained. His current work is on generalizations of group convolutional NNs, improvements of their computational and representation efficiency through sparse (graphs) and adaptive learning mechanisms. Erik is a recipient of a MICCAI Young Scientist Award 2018, Philips Impact Award (MIDL 2018) and a personal VENI research grant (awarded by the Dutch Research Council (NWO)).

Curriculum vitae (16 nov 2022)
PhD thesis (8 MB) or (48 MB)

Research topics

Neural Ideograms
Group equivariant deep learning
Leveraging geometry in (graph) neural networks
Computing with geometric quantities (manifold-valued features)
Geometric latent space modeling

Team

David Romero: Continuous kernel CNNs, Group equivariant networks and Self-attention networks.
Putri van der Linden: Geometric Deep Learning, Sparse Visual Representations
Rob Hesselink: Geometric Deep Learning, Graphs and Group Equivariance
Sharvaree Vadgama: Geometric Latent Space Modeling and Explainable AI
Michel Botros: AI-Based Detection and Prediction of Esophageal Cancer
Mohammad Islam: Uncertainty-aware AI-based Predictive Modeling for Treatment Decision Making

Selected Publications

NeurREPS @ NeurIPS

Kendall Shape-VAE : Learning Shapes in a Generative Framework

Vadgama, Sharvaree, Tomczak, Jakub Mikolaj, and Bekkers, Erik J

In NeurIPS 2022 Workshop on Symmetry and Geometry in Neural Representations 2022

Abs HTML PDF

Learning an interpretable representation of data without supervision is an important precursor for the development of artificial intelligence. In this work, we introduce \textitKendall Shape-VAE, a novel Variational Autoencoder framework for learning shapes as it disentangles the latent space by compressing information to simpler geometric symbols. In \textitKendall Shape-VAE, we modify the Hyperspherical Variational Autoencoder such that it results in an exactly rotationally equivariant network using the notion of landmarks in the Kendall shape space. We show the exact equivariance of the model through experiments on rotated MNIST.
ICML

Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups

Knigge, David M, Romero, David W, and Bekkers, Erik J

International Conference on Machine Learning 2022

Abs HTML PDF

Group convolutional neural networks (G-CNNs) have been shown to increase parameter efficiency and model accuracy by incorporating geometric inductive biases. In this work, we investigate the properties of representations learned by regular G-CNNs, and show considerable parameter redundancy in group convolution kernels. This finding motivates further weight-tying by sharing convolution kernels over subgroups. To this end, we introduce convolution kernels that are separable over the subgroup and channel dimensions. In order to obtain equivariance to arbitrary affine Lie groups we provide a continuous parameterisation of separable convolution kernels. We evaluate our approach across several vision datasets, and show that our weight sharing leads to improved performance and computational efficiency. In many settings, separable G-CNNs outperform their non-separable counterpart, while only using a fraction of their training time. In addition, thanks to the increase in computational efficiency, we are able to implement G-CNNs equivariant to the Sim(2) group; the group of dilations, rotations and translations. Sim(2)-equivariance further improves performance on all tasks considered.
ICLR

Geometric and Physical Quantities improve E (3) Equivariant Message Passing

Brandstetter, Johannes, Hesselink, Rob, Pol, Elise, Bekkers, Erik, and Welling, Max

In International Conference on Learning Representations 2022

Abs HTML PDF Code

Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLPs provide a new class of activation functions for general use with steerable feature fields. We discuss ours and related work through the lens of equivariant non-linear convolutions, which further allows us to pin-point the successful components of SEGNNs: non-linear message aggregation improves upon classic linear (steerable) point convolutions; steerable messages improve upon recent equivariant graph networks that send invariant messages. We demonstrate the effectiveness of our method on several tasks in computational physics and chemistry and provide extensive ablation studies.
ICLR

B-Spline CNNs on Lie groups

Bekkers, Erik J

In International Conference on Learning Representations 2019

Abs HTML PDF Code Video

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 (PCam dataset) in which rotation equivariance plays a key role and facial landmark localization (CelebA dataset) 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.