Erik J Bekkers

Erik J Bekkers, Sharvaree Vadgama, Rob D Hesselink et al.

Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions $\mathbb{R}^3$, position and orientations $\mathbb{R}^3 {\times} S^2$, and the group $SE(3)$ itself. Among these, $\mathbb{R}^3 {\times} S^2$ is an optimal choice due to the ability to represent directional information, which $\mathbb{R}^3$ methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full $SE(3)$ group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

David R Wessels, David M Knigge, Samuele Papa et al.

Conditional Neural Fields (CNFs) are increasingly being leveraged as continuous signal representations, by associating each data-sample with a latent variable that conditions a shared backbone Neural Field (NeF) to reconstruct the sample. However, existing CNF architectures face limitations when using this latent downstream in tasks requiring fine-grained geometric reasoning, such as classification and segmentation. We posit that this results from lack of explicit modelling of geometric information (e.g., locality in the signal or the orientation of a feature) in the latent space of CNFs. As such, we propose Equivariant Neural Fields (ENFs), a novel CNF architecture which uses a geometry-informed cross-attention to condition the NeF on a geometric variable--a latent point cloud of features--that enables an equivariant decoding from latent to field. We show that this approach induces a steerability property by which both field and latent are grounded in geometry and amenable to transformation laws: if the field transforms, the latent representation transforms accordingly--and vice versa. Crucially, this equivariance relation ensures that the latent is capable of (1) representing geometric patterns faithfully, allowing for geometric reasoning in latent space, and (2) weight-sharing over similar local patterns, allowing for efficient learning of datasets of fields. We validate these main properties in a range of tasks including classification, segmentation, forecasting, reconstruction and generative modelling, showing clear improvement over baselines with a geometry-free latent space. Code attached to submission https://github.com/Dafidofff/enf-jax. Code for a clean and minimal repo https://github.com/david-knigge/enf-min-jax.

Mohammad Mohaiminul Islam, Coen de Vente, Bart Liefers et al.

In this paper, we present a new approach for uncertainty-aware retinal layer segmentation in Optical Coherence Tomography (OCT) scans using probabilistic signed distance functions (SDF). Traditional pixel-wise and regression-based methods primarily encounter difficulties in precise segmentation and lack of geometrical grounding respectively. To address these shortcomings, our methodology refines the segmentation by predicting a signed distance function (SDF) that effectively parameterizes the retinal layer shape via level set. We further enhance the framework by integrating probabilistic modeling, applying Gaussian distributions to encapsulate the uncertainty in the shape parameterization. This ensures a robust representation of the retinal layer morphology even in the presence of ambiguous input, imaging noise, and unreliable segmentations. Both quantitative and qualitative evaluations demonstrate superior performance when compared to other methods. Additionally, we conducted experiments on artificially distorted datasets with various noise types-shadowing, blinking, speckle, and motion-common in OCT scans to showcase the effectiveness of our uncertainty estimation. Our findings demonstrate the possibility to obtain reliable segmentation of retinal layers, as well as an initial step towards the characterization of layer integrity, a key biomarker for disease progression. Our code is available at \url{https://github.com/niazoys/RLS_PSDF}.

Floor Eijkelboom, Heiko Zimmermann, Sharvaree Vadgama et al.

We derive a controlled generation objective within the framework of Variational Flow Matching (VFM), which casts flow matching as a variational inference problem. We demonstrate that controlled generation can be implemented two ways: (1) by way of end-to-end training of conditional generative models, or (2) as a Bayesian inference problem, enabling post hoc control of unconditional models without retraining. Furthermore, we establish the conditions required for equivariant generation and provide an equivariant formulation of VFM tailored for molecular generation, ensuring invariance to rotations, translations, and permutations. We evaluate our approach on both uncontrolled and controlled molecular generation, achieving state-of-the-art performance on uncontrolled generation and outperforming state-of-the-art models in controlled generation, both with end-to-end training and in the Bayesian inference setting. This work strengthens the connection between flow-based generative modeling and Bayesian inference, offering a scalable and principled framework for constraint-driven and symmetry-aware generation.

Erik J Bekkers, Anna Ciaunica

Imagine an Artificial Intelligence (AI) that perfectly mimics human emotion and begs for its continued existence. Is it morally permissible to unplug it? What if limited resources force a choice between unplugging such a pleading AI or a silent pre-term infant? We term this the unplugging paradox. This paper critically examines the deeply ingrained physicalist assumptions-specifically computational functionalism-that keep this dilemma afloat. We introduce Biological Idealism, a framework that-unlike physicalism-remains logically coherent and empirically consistent. In this view, conscious experiences are fundamental and autopoietic life its necessary physical signature. This yields a definitive conclusion: AI is at best a functional mimic, not a conscious experiencing subject. We discuss how current AI consciousness theories erode moral standing criteria, and urge a shift from speculative machine rights to protecting human conscious life. The real moral issue lies not in making AI conscious and afraid of death, but in avoiding transforming humans into zombies.

Aniss Aiman Medbouhi, Alejandro García-Castellanos, Giovanni Luca Marchetti et al.

We study the problem of constructing Steiner Minimal Trees (SMTs) in hyperbolic space. Exact SMT computation is NP-hard, and existing hyperbolic heuristics such as HyperSteiner are deterministic and often get trapped in locally suboptimal configurations. We introduce Randomized HyperSteiner (RHS), a stochastic Delaunay triangulation heuristic that incorporates randomness into the expansion process and refines candidate trees via Riemannian gradient descent optimization. Experiments on synthetic data sets and a real-world single-cell transcriptomic data show that RHS outperforms Minimum Spanning Tree (MST), Neighbour Joining, and vanilla HyperSteiner (HS). In near-boundary configurations, RHS can achieve a 32% reduction in total length over HS, demonstrating its effectiveness and robustness in diverse data regimes.

Johannes Brandstetter, Rob Hesselink, Elise van der Pol et al.

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.

Erik J Bekkers

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 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.

Erik J Bekkers, Maxime W Lafarge, Mitko Veta et al.

We propose a framework for rotation and translation covariant deep learning using $SE(2)$ group convolutions. The group product of the special Euclidean motion group $SE(2)$ describes how a concatenation of two roto-translations results in a net roto-translation. We encode this geometric structure into convolutional neural networks (CNNs) via $SE(2)$ group convolutional layers, which fit into the standard 2D CNN framework, and which allow to generically deal with rotated input samples without the need for data augmentation. We introduce three layers: a lifting layer which lifts a 2D (vector valued) image to an $SE(2)$-image, i.e., 3D (vector valued) data whose domain is $SE(2)$; a group convolution layer from and to an $SE(2)$-image; and a projection layer from an $SE(2)$-image to a 2D image. The lifting and group convolution layers are $SE(2)$ covariant (the output roto-translates with the input). The final projection layer, a maximum intensity projection over rotations, makes the full CNN rotation invariant. We show with three different problems in histopathology, retinal imaging, and electron microscopy that with the proposed group CNNs, state-of-the-art performance can be achieved, without the need for data augmentation by rotation and with increased performance compared to standard CNNs that do rely on augmentation.