Erik J. Bekkers

Putri A. van der Linden, David W. Romero, Erik J. Bekkers

Neural operations that rely on neighborhood information are much more expensive when deployed on point clouds than on grid data due to the irregular distances between points in a point cloud. In a grid, on the other hand, we can compute the kernel only once and reuse it for all query positions. As a result, operations that rely on neighborhood information scale much worse for point clouds than for grid data, specially for large inputs and large neighborhoods. In this work, we address the scalability issue of point cloud methods by tackling its root cause: the irregularity of the data. We propose learnable gridification as the first step in a point cloud processing pipeline to transform the point cloud into a compact, regular grid. Thanks to gridification, subsequent layers can use operations defined on regular grids, e.g., Conv3D, which scale much better than native point cloud methods. We then extend gridification to point cloud to point cloud tasks, e.g., segmentation, by adding a learnable de-gridification step at the end of the point cloud processing pipeline to map the compact, regular grid back to its original point cloud form. Through theoretical and empirical analysis, we show that gridified networks scale better in terms of memory and time than networks directly applied on raw point cloud data, while being able to achieve competitive results. Our code is publicly available at https://github.com/computri/gridifier.

David M. Knigge, David W. Romero, Albert Gu et al.

Performant Convolutional Neural Network (CNN) architectures must be tailored to specific tasks in order to consider the length, resolution, and dimensionality of the input data. In this work, we tackle the need for problem-specific CNN architectures. We present the Continuous Convolutional Neural Network (CCNN): a single CNN able to process data of arbitrary resolution, dimensionality and length without any structural changes. Its key component are its continuous convolutional kernels which model long-range dependencies at every layer, and thus remove the need of current CNN architectures for task-dependent downsampling and depths. We showcase the generality of our method by using the same architecture for tasks on sequential ($1{\rm D}$), visual ($2{\rm D}$) and point-cloud ($3{\rm D}$) data. Our CCNN matches and often outperforms the current state-of-the-art across all tasks considered.

Artem Moskalev, Anna Sepliarskaia, Erik J. Bekkers et al.

In this paper, we investigate properties and limitations of invariance learned by neural networks from the data compared to the genuine invariance achieved through invariant weight-tying. To do so, we adopt a group theoretical perspective and analyze invariance learning in neural networks without weight-tying constraints. We demonstrate that even when a network learns to correctly classify samples on a group orbit, the underlying decision-making in such a model does not attain genuine invariance. Instead, learned invariance is strongly conditioned on the input data, rendering it unreliable if the input distribution shifts. We next demonstrate how to guide invariance learning toward genuine invariance by regularizing the invariance of a model at the training. To this end, we propose several metrics to quantify learned invariance: (i) predictive distribution invariance, (ii) logit invariance, and (iii) saliency invariance similarity. We show that the invariance learned with the invariance error regularization closely reassembles the genuine invariance of weight-tying models and reliably holds even under a severe input distribution shift. Closer analysis of the learned invariance also reveals the spectral decay phenomenon, when a network chooses to achieve the invariance to a specific transformation group by reducing the sensitivity to any input perturbation.

David W. Romero, David M. Knigge, Albert Gu et al.

The use of Convolutional Neural Networks (CNNs) is widespread in Deep Learning due to a range of desirable model properties which result in an efficient and effective machine learning framework. However, performant CNN architectures must be tailored to specific tasks in order to incorporate considerations such as the input length, resolution, and dimentionality. In this work, we overcome the need for problem-specific CNN architectures with our Continuous Convolutional Neural Network (CCNN): a single CNN architecture equipped with continuous convolutional kernels that can be used for tasks on data of arbitrary resolution, dimensionality and length without structural changes. Continuous convolutional kernels model long range dependencies at every layer, and remove the need for downsampling layers and task-dependent depths needed in current CNN architectures. We show the generality of our approach by applying the same CCNN to a wide set of tasks on sequential (1$\mathrm{D}$) and visual data (2$\mathrm{D}$). Our CCNN performs competitively and often outperforms the current state-of-the-art across all tasks considered.

Miltiadis Kofinas, Erik J. Bekkers, Naveen Shankar Nagaraja et al.

Systems of interacting objects often evolve under the influence of field effects that govern their dynamics, yet previous works have abstracted away from such effects, and assume that systems evolve in a vacuum. In this work, we focus on discovering these fields, and infer them from the observed dynamics alone, without directly observing them. We theorize the presence of latent force fields, and propose neural fields to learn them. Since the observed dynamics constitute the net effect of local object interactions and global field effects, recently popularized equivariant networks are inapplicable, as they fail to capture global information. To address this, we propose to disentangle local object interactions -- which are $\mathrm{SE}(n)$ equivariant and depend on relative states -- from external global field effects -- which depend on absolute states. We model interactions with equivariant graph networks, and combine them with neural fields in a novel graph network that integrates field forces. Our experiments show that we can accurately discover the underlying fields in charged particles settings, traffic scenes, and gravitational n-body problems, and effectively use them to learn the system and forecast future trajectories.

Thijs P. Kuipers, Erik J. Bekkers

Regular group convolutional neural networks (G-CNNs) have been shown to increase model performance and improve equivariance to different geometrical symmetries. This work addresses the problem of SE(3), i.e., roto-translation equivariance, on volumetric data. Volumetric image data is prevalent in many medical settings. Motivated by the recent work on separable group convolutions, we devise a SE(3) group convolution kernel separated into a continuous SO(3) (rotation) kernel and a spatial kernel. We approximate equivariance to the continuous setting by sampling uniform SO(3) grids. Our continuous SO(3) kernel is parameterized via RBF interpolation on similarly uniform grids. We demonstrate the advantages of our approach in volumetric medical image analysis. Our SE(3) equivariant models consistently outperform CNNs and regular discrete G-CNNs on challenging medical classification tasks and show significantly improved generalization capabilities. Our approach achieves up to a 16.5% gain in accuracy over regular CNNs.

Oline Ranum, David R. Wessels, Gomer Otterspeer et al.

Sign Language Processing (SLP) provides a foundation for a more inclusive future in language technology; however, the field faces several significant challenges that must be addressed to achieve practical, real-world applications. This work addresses multi-view isolated sign recognition (MV-ISR), and highlights the essential role of 3D awareness and geometry in SLP systems. We introduce the NGT200 dataset, a novel spatio-temporal multi-view benchmark, establishing MV-ISR as distinct from single-view ISR (SV-ISR). We demonstrate the benefits of synthetic data and propose conditioning sign representations on spatial symmetries inherent in sign language. Leveraging an SE(2) equivariant model improves MV-ISR performance by 8%-22% over the baseline.

Soufiane Ben Haddou, Laura Alvarez-Florez, Erik J. Bekkers et al.

Late gadolinium enhancement (LGE) imaging is the clinical standard for myocardial scar assessment, but limited annotated datasets hinder the development of automated segmentation methods. We propose a novel framework that synthesises both LGE images and their corresponding segmentation masks using implicit neural representations (INRs) combined with denoising diffusion models. Our approach first trains INRs to capture continuous spatial representations of LGE data and associated myocardium and fibrosis masks. These INRs are then compressed into compact latent embeddings, preserving essential anatomical information. A diffusion model operates on this latent space to generate new representations, which are decoded into synthetic LGE images with anatomically consistent segmentation masks. Experiments on 133 cardiac MRI scans suggest that augmenting training data with 200 synthetic volumes contributes to improved fibrosis segmentation performance, with the Dice score showing an increase from 0.509 to 0.524. Our approach provides an annotation-free method to help mitigate data scarcity.The code for this research is publicly available.

Olga Zaghen, Maksim Zhdanov, Dario Coscia et al.

Machine Learning Interatomic Potentials (MLIPs) achieve near ab initio accuracy at a fraction of the cost of quantum-mechanical simulations, yet they remain prone to silent failures on out-of-distribution configurations, making principled uncertainty quantification (UQ) essential for error-aware simulations and active learning. Existing non-ensemble UQ methods for MLIPs rely either on variational inference or on parametric distributional assumptions, both of which add architectural complexity and hyper-parameters that must be tuned per task. Inspired by recent advances in probabilistic weather forecasting, we propose a simpler alternative: turn a deterministic MLIP into a probabilistic one through learned functional perturbations and finetune it end-to-end with the Continuous Ranked Probability Score (CRPS), a proper scoring rule. We validate the approach with an equivariant GNN (P-EGNN) trained from scratch and by finetuning the foundation model the Orb-v3 for silica. On the N-body charged particle benchmark, P-EGNN improves CRPS over the state-of-the-art Bayesian MLIP method BLIP by 19-32% across all training sizes; on silica, P-Orb raises the Spearman correlation between predicted uncertainty and actual error from 0.75 (BLIP-Orb) to 0.84.

Vivien van Veldhuizen, Sharvaree Vadgama, Onno J. de Boer et al.

Early detection of Barrett's Esophagus (BE), the only known precursor to Esophageal adenocarcinoma (EAC), is crucial for effectively preventing and treating esophageal cancer. In this work, we investigate the potential of geometric Variational Autoencoders (VAEs) to learn a meaningful latent representation that captures the progression of BE. We show that hyperspherical VAE (S-VAE) and Kendall Shape VAE show improved classification accuracy, reconstruction loss, and generative capacity. Additionally, we present a novel autoencoder architecture that can generate qualitative images without the need for a variational framework while retaining the benefits of an autoencoder, such as improved stability and reconstruction quality.

Maria Esteban-Casadevall, Jorge Carrasco-Pollo, Max Welling et al.

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation.

Laura Alvarez-Florez, Angel Bujalance-Gomez, Femke Raijmakers et al.

Cardiac magnetic resonance imaging (CMR) offers detailed evaluation of cardiac structure and function, but its limited accessibility restricts use to selected patient populations. In contrast, the electrocardiogram (ECG) is ubiquitous and inexpensive, and provides rich information on cardiac electrical activity and rhythm, yet offers limited insight into underlying cardiac structure and mechanical function. To address this, we introduce a contrastive learning framework that improves the extraction of clinically relevant cardiac phenotypes from ECG by learning from paired ECG-CMR data. Our approach aligns ECG representations with 3D CMR volumes at end-diastole (ED) and end-systole (ES), with a dual-phase contrastive loss to anchor each ECG jointly with both cardiac phases in a shared latent space. Unlike prior methods limited to 2D CMR representations with or without a temporal component, our framework models 3D anatomy at both ED and ES phases as distinct latent representations, enabling flexible disentanglement of structural and functional cardiac properties. Using over 34,000 ECG-CMR pairs from the UK Biobank, we demonstrate improved extraction of image-derived phenotypes from ECG, particularly for functional parameters ($\uparrow$ 9.2\%), while improvements in clinical outcome prediction remained modest ($\uparrow$ 0.7\%). This strategy could enable scalable and cost-effective extraction of image-derived traits from ECG. The code for this research is publicly available.

David W. Romero, Erik J. Bekkers, Jakub M. Tomczak et al.

Leveraging the symmetries inherent to specific data domains for the construction of equivariant neural networks has lead to remarkable improvements in terms of data efficiency and generalization. However, most existing research focuses on symmetries arising from planar and volumetric data, leaving a crucial data source largely underexplored: time-series. In this work, we fill this gap by leveraging the symmetries inherent to time-series for the construction of equivariant neural network. We identify two core symmetries: *scale and translation*, and construct scale-translation equivariant neural networks for time-series learning. Intriguingly, we find that scale-translation equivariant mappings share strong resemblance with the wavelet transform. Inspired by this resemblance, we term our networks Wavelet Networks, and show that they perform nested non-linear wavelet-like time-frequency transforms. Empirical results show that Wavelet Networks outperform conventional CNNs on raw waveforms, and match strongly engineered spectrogram techniques across several tasks and time-series types, including audio, environmental sounds, and electrical signals. Our code is publicly available at https://github.com/dwromero/wavelet_networks.

Putri A. van der Linden, Alejandro García-Castellanos, Sharvaree Vadgama et al.

Group equivariance has emerged as a valuable inductive bias in deep learning, enhancing generalization, data efficiency, and robustness. Classically, group equivariant methods require the groups of interest to be known beforehand, which may not be realistic for real-world data. Additionally, baking in fixed group equivariance may impose overly restrictive constraints on model architecture. This highlights the need for methods that can dynamically discover and apply symmetries as soft constraints. For neural network architectures, equivariance is commonly achieved through group transformations of a canonical weight tensor, resulting in weight sharing over a given group $G$. In this work, we propose to learn such a weight-sharing scheme by defining a collection of learnable doubly stochastic matrices that act as soft permutation matrices on canonical weight tensors, which can take regular group representations as a special case. This yields learnable kernel transformations that are jointly optimized with downstream tasks. We show that when the dataset exhibits strong symmetries, the permutation matrices will converge to regular group representations and our weight-sharing networks effectively become regular group convolutions. Additionally, the flexibility of the method enables it to effectively pick up on partial symmetries.

Olga Zaghen, Floor Eijkelboom, Alison Pouplin et al.

We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. In Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) are largely equivalent due to affine interpolations. On curved manifolds this equivalence breaks down, and we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. Building on this insight, we derive a variational flow matching objective based on Riemannian Gaussian distributions, applicable to manifolds with closed-form geodesics. We formally analyze its relationship to Riemannian Flow Matching (RFM), exposing that the RFM objective lacks a curvature-dependent penalty - encoded via Jacobi fields - that is naturally present in RG-VFM. Experiments on synthetic spherical and hyperbolic benchmarks, as well as real-world tasks in material and protein generation, demonstrate that RG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity-based baselines.

Mohammad Mohaiminul Islam, Rishabh Anand, David R. Wessels et al.

While widespread, Transformers lack inductive biases for geometric symmetries common in science and computer vision. Existing equivariant methods often sacrifice the efficiency and flexibility that make Transformers so effective through complex, computationally intensive designs. We introduce the Platonic Transformer to resolve this trade-off. By defining attention relative to reference frames from the Platonic solid symmetry groups, our method induces a principled weight-sharing scheme. This enables combined equivariance to continuous translations and Platonic symmetries, while preserving the exact architecture and computational cost of a standard Transformer. Furthermore, we show that this attention is formally equivalent to a dynamic group convolution, which reveals that the model learns adaptive geometric filters and enables a highly scalable, linear-time convolutional variant. Across diverse benchmarks in computer vision (CIFAR-10), 3D point clouds (ScanObjectNN), and molecular property prediction (QM9, OMol25), the Platonic Transformer achieves competitive performance by leveraging these geometric constraints at no additional cost.

Putri A. van der Linden, Alexander Timans, Dharmesh Tailor et al.

Equivariant models leverage prior knowledge on symmetries to improve predictive performance, but misspecified architectural constraints can harm it instead. While work has explored learning or relaxing constraints, selecting among pretrained models with varying symmetry biases remains challenging. We examine this model selection task from an uncertainty-aware perspective, comparing frequentist (via Conformal Prediction), Bayesian (via the marginal likelihood), and calibration-based measures to naive error-based evaluation. We find that uncertainty metrics generally align with predictive performance, but Bayesian model evidence does so inconsistently. We attribute this to a mismatch in Bayesian and geometric notions of model complexity for the employed last-layer Laplace approximation, and discuss possible remedies. Our findings point towards the potential of uncertainty in guiding symmetry-aware model selection.

Putri A. van der Linden, Alexander Timans, Erik J. Bekkers

We study the problem of conformal prediction (CP) under geometric data shifts, where data samples are susceptible to transformations such as rotations or flips. While CP endows prediction models with post-hoc uncertainty quantification and formal coverage guarantees, their practicality breaks under distribution shifts that deteriorate model performance. To address this issue, we propose integrating geometric information--such as geometric pose--into the conformal procedure to reinstate its guarantees and ensure robustness under geometric shifts. In particular, we explore recent advancements on pose canonicalization as a suitable information extractor for this purpose. Evaluating the combined approach across discrete and continuous shifts and against equivariant and augmentation-based baselines, we find that integrating geometric information with CP yields a principled way to address geometric shifts while maintaining broad applicability to black-box predictors.

Finn M. Sherry, Chase van de Geijn, Erik J. Bekkers et al.

We axiomatically derive a family of wavelets for an orientation score, lifting from position space $\mathbb{R}^2$ to position and orientation space $\mathbb{R}^2\times S^1$, with fast reconstruction property, that minimise position-orientation uncertainty. We subsequently show that these minimum uncertainty states are well-approximated by cake wavelets: for standard parameters, the uncertainty gap of cake wavelets is less than 1.1, and in the limit, we prove the uncertainty gap tends to the minimum of 1. Next, we complete a previous theoretical argument that one does not have to train the lifting layer in (PDE-)G-CNNs, but can instead use cake wavelets. Finally, we show experimentally that in this way we can reduce the network complexity and improve the interpretability of (PDE-)G-CNNs, with only a slight impact on the model's performance.

Mohammad Mohaiminul Islam, Thijs P. Kuipers, Sharvaree Vadgama et al.

Generative models for sequential data often struggle with sparsely sampled and high-dimensional trajectories, typically reducing the learning of dynamics to pairwise transitions. We propose Interpolative Multi-Marginal Flow Matching (IMMFM), a framework that learns continuous stochastic dynamics jointly consistent with multiple observed time points. IMMFM employs a piecewise-quadratic interpolation path as a smooth target for flow matching and jointly optimizes drift and a data-driven diffusion coefficient, supported by a theoretical condition for stable learning. This design captures intrinsic stochasticity, handles irregular sparse sampling, and yields subject-specific trajectories. Experiments on synthetic benchmarks and real-world longitudinal neuroimaging datasets show that IMMFM outperforms existing methods in both forecasting accuracy and further downstream tasks.

Alejandro García-Castellanos, David R. Wessels, Nicky J. van den Berg et al.

We introduce Equivariant Neural Eikonal Solvers, a novel framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. Our approach employs a single neural field where a unified shared backbone is conditioned on signal-specific latent variables - represented as point clouds in a Lie group - to model diverse Eikonal solutions. The ENF integration ensures equivariant mapping from these latent representations to the solution field, delivering three key benefits: enhanced representation efficiency through weight-sharing, robust geometric grounding, and solution steerability. This steerability allows transformations applied to the latent point cloud to induce predictable, geometrically meaningful modifications in the resulting Eikonal solution. By coupling these steerable representations with Physics-Informed Neural Networks (PINNs), our framework accurately models Eikonal travel-time solutions while generalizing to arbitrary Riemannian manifolds with regular group actions. This includes homogeneous spaces such as Euclidean, position-orientation, spherical, and hyperbolic manifolds. We validate our approach through applications in seismic travel-time modeling of 2D, 3D, and spherical benchmark datasets. Experimental results demonstrate superior performance, scalability, adaptability, and user controllability compared to existing Neural Operator-based Eikonal solver methods.

Andrew Draganov, Sharvaree Vadgama, Erik J. Bekkers

We show that the gradient of the cosine similarity between two points goes to zero in two under-explored settings: (1) if a point has large magnitude or (2) if the points are on opposite ends of the latent space. Counterintuitively, we prove that optimizing the cosine similarity between points forces them to grow in magnitude. Thus, (1) is unavoidable in practice. We then observe that these derivations are extremely general -- they hold across deep learning architectures and for many of the standard self-supervised learning (SSL) loss functions. This leads us to propose cut-initialization: a simple change to network initialization that helps all studied SSL methods converge faster.

David M. Knigge, David R. Wessels, Riccardo Valperga et al.

Recently, Conditional Neural Fields (NeFs) have emerged as a powerful modelling paradigm for PDEs, by learning solutions as flows in the latent space of the Conditional NeF. Although benefiting from favourable properties of NeFs such as grid-agnosticity and space-time-continuous dynamics modelling, this approach limits the ability to impose known constraints of the PDE on the solutions -- e.g. symmetries or boundary conditions -- in favour of modelling flexibility. Instead, we propose a space-time continuous NeF-based solving framework that - by preserving geometric information in the latent space - respects known symmetries of the PDE. We show that modelling solutions as flows of pointclouds over the group of interest $G$ improves generalization and data-efficiency. We validated that our framework readily generalizes to unseen spatial and temporal locations, as well as geometric transformations of the initial conditions - where other NeF-based PDE forecasting methods fail - and improve over baselines in a number of challenging geometries.

Veljko Kovač, Erik J. Bekkers, Pietro Liò et al.

This paper introduces E(n) Equivariant Message Passing Cellular Networks (EMPCNs), an extension of E(n) Equivariant Graph Neural Networks to CW-complexes. Our approach addresses two aspects of geometric message passing networks: 1) enhancing their expressiveness by incorporating arbitrary cells, and 2) achieving this in a computationally efficient way with a decoupled EMPCNs technique. We demonstrate that EMPCNs achieve close to state-of-the-art performance on multiple tasks without the need for steerability, including many-body predictions and motion capture. Moreover, ablation studies confirm that decoupled EMPCNs exhibit stronger generalization capabilities than their non-topologically informed counterparts. These findings show that EMPCNs can be used as a scalable and expressive framework for higher-order message passing in geometric and topological graphs

Yeskendir Koishekenov, Erik J. Bekkers

The flexibility and effectiveness of message passing based graph neural networks (GNNs) induced considerable advances in deep learning on graph-structured data. In such approaches, GNNs recursively update node representations based on their neighbors and they gain expressivity through the use of node and edge attribute vectors. E.g., in computational tasks such as physics and chemistry usage of edge attributes such as relative position or distance proved to be essential. In this work, we address not what kind of attributes to use, but how to condition on this information to improve model performance. We consider three types of conditioning; weak, strong, and pure, which respectively relate to concatenation-based conditioning, gating, and transformations that are causally dependent on the attributes. This categorization provides a unifying viewpoint on different classes of GNNs, from separable convolutions to various forms of message passing networks. We provide an empirical study on the effect of conditioning methods in several tasks in computational chemistry.

Hugo Aguettaz, Erik J. Bekkers, Michaël Defferrard

We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds. Surfing on the success of graph- and group-based neural networks, we take advantage of the recent developments in the geometric deep learning field to derive a new approach to exploit any anisotropies in data. Via discrete approximations of Lie groups, we develop a graph neural network made of anisotropic convolutional layers (Chebyshev convolutions), spatial pooling and unpooling layers, and global pooling layers. Group equivariance is achieved via equivariant and invariant operators on graphs with anisotropic left-invariant Riemannian distance-based affinities encoded on the edges. Thanks to its simple form, the Riemannian metric can model any anisotropies, both in the spatial and orientation domains. This control on anisotropies of the Riemannian metrics allows to balance equivariance (anisotropic metric) against invariance (isotropic metric) of the graph convolution layers. Hence we open the doors to a better understanding of anisotropic properties. Furthermore, we empirically prove the existence of (data-dependent) sweet spots for anisotropic parameters on CIFAR10. This crucial result is evidence of the benefice we could get by exploiting anisotropic properties in data. We also evaluate the scalability of this approach on STL10 (image data) and ClimateNet (spherical data), showing its remarkable adaptability to diverse tasks.

Jan Zuiderveld, Marco Federici, Erik J. Bekkers

The high temporal resolution of audio and our perceptual sensitivity to small irregularities in waveforms make synthesizing at high sampling rates a complex and computationally intensive task, prohibiting real-time, controllable synthesis within many approaches. In this work we aim to shed light on the potential of Conditional Implicit Neural Representations (CINRs) as lightweight backbones in generative frameworks for audio synthesis. Our experiments show that small Periodic Conditional INRs (PCINRs) learn faster and generally produce quantitatively better audio reconstructions than Transposed Convolutional Neural Networks with equal parameter counts. However, their performance is very sensitive to activation scaling hyperparameters. When learning to represent more uniform sets, PCINRs tend to introduce artificial high-frequency components in reconstructions. We validate this noise can be minimized by applying standard weight regularization during training or decreasing the compositional depth of PCINRs, and suggest directions for future research.

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

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 $\mathrm{Sim(2)}$ group; the group of dilations, rotations and translations. $\mathrm{Sim(2)}$-equivariance further improves performance on all tasks considered.

David W. Romero, Robert-Jan Bruintjes, Jakub M. Tomczak et al.

When designing Convolutional Neural Networks (CNNs), one must select the size\break of the convolutional kernels before training. Recent works show CNNs benefit from different kernel sizes at different layers, but exploring all possible combinations is unfeasible in practice. A more efficient approach is to learn the kernel size during training. However, existing works that learn the kernel size have a limited bandwidth. These approaches scale kernels by dilation, and thus the detail they can describe is limited. In this work, we propose FlexConv, a novel convolutional operation with which high bandwidth convolutional kernels of learnable kernel size can be learned at a fixed parameter cost. FlexNets model long-term dependencies without the use of pooling, achieve state-of-the-art performance on several sequential datasets, outperform recent works with learned kernel sizes, and are competitive with much deeper ResNets on image benchmark datasets. Additionally, FlexNets can be deployed at higher resolutions than those seen during training. To avoid aliasing, we propose a novel kernel parameterization with which the frequency of the kernels can be analytically controlled. Our novel kernel parameterization shows higher descriptive power and faster convergence speed than existing parameterizations. This leads to important improvements in classification accuracy.

David W. Romero, Anna Kuzina, Erik J. Bekkers et al.

Conventional neural architectures for sequential data present important limitations. Recurrent networks suffer from exploding and vanishing gradients, small effective memory horizons, and must be trained sequentially. Convolutional networks are unable to handle sequences of unknown size and their memory horizon must be defined a priori. In this work, we show that all these problems can be solved by formulating convolutional kernels in CNNs as continuous functions. The resulting Continuous Kernel Convolution (CKConv) allows us to model arbitrarily long sequences in a parallel manner, within a single operation, and without relying on any form of recurrence. We show that Continuous Kernel Convolutional Networks (CKCNNs) obtain state-of-the-art results in multiple datasets, e.g., permuted MNIST, and, thanks to their continuous nature, are able to handle non-uniformly sampled datasets and irregularly-sampled data natively. CKCNNs match or perform better than neural ODEs designed for these purposes in a faster and simpler manner.

Maxime W. Lafarge, Erik J. Bekkers, Josien P. W. Pluim et al.

Rotation-invariance is a desired property of machine-learning models for medical image analysis and in particular for computational pathology applications. We propose a framework to encode the geometric structure of the special Euclidean motion group SE(2) in convolutional networks to yield translation and rotation equivariance via the introduction of SE(2)-group convolution layers. This structure enables models to learn feature representations with a discretized orientation dimension that guarantees that their outputs are invariant under a discrete set of rotations. Conventional approaches for rotation invariance rely mostly on data augmentation, but this does not guarantee the robustness of the output when the input is rotated. At that, trained conventional CNNs may require test-time rotation augmentation to reach their full capability. This study is focused on histopathology image analysis applications for which it is desirable that the arbitrary global orientation information of the imaged tissues is not captured by the machine learning models. The proposed framework is evaluated on three different histopathology image analysis tasks (mitosis detection, nuclei segmentation and tumor classification). We present a comparative analysis for each problem and show that consistent increase of performances can be achieved when using the proposed framework.

David W. Romero, Erik J. Bekkers, Jakub M. Tomczak et al.

Although group convolutional networks are able to learn powerful representations based on symmetry patterns, they lack explicit means to learn meaningful relationships among them (e.g., relative positions and poses). In this paper, we present attentive group equivariant convolutions, a generalization of the group convolution, in which attention is applied during the course of convolution to accentuate meaningful symmetry combinations and suppress non-plausible, misleading ones. We indicate that prior work on visual attention can be described as special cases of our proposed framework and show empirically that our attentive group equivariant convolutional networks consistently outperform conventional group convolutional networks on benchmark image datasets. Simultaneously, we provide interpretability to the learned concepts through the visualization of equivariant attention maps.

Erik J. Bekkers, Marco Loog, Bart M. ter Haar Romeny et al.

We propose a template matching method for the detection of 2D image objects that are characterized by orientation patterns. Our method is based on data representations via orientation scores, which are functions on the space of positions and orientations, and which are obtained via a wavelet-type transform. This new representation allows us to detect orientation patterns in an intuitive and direct way, namely via cross-correlations. Additionally, we propose a generalized linear regression framework for the construction of suitable templates using smoothing splines. Here, it is important to recognize a curved geometry on the position-orientation domain, which we identify with the Lie group SE(2): the roto-translation group. Templates are then optimized in a B-spline basis, and smoothness is defined with respect to the curved geometry. We achieve state-of-the-art results on three different applications: detection of the optic nerve head in the retina (99.83% success rate on 1737 images), of the fovea in the retina (99.32% success rate on 1616 images), and of the pupil in regular camera images (95.86% on 1521 images). The high performance is due to inclusion of both intensity and orientation features with effective geometric priors in the template matching. Moreover, our method is fast due to a cross-correlation based matching approach.