Floor Eijkelboom

Floor Eijkelboom, Erik Bekkers, Michael Bronstein et al.

The most prevalent class of neural networks operating on graphs are message passing neural networks (MPNNs), in which the representation of a node is updated iteratively by aggregating information in the 1-hop neighborhood. Since this paradigm for computing node embeddings may prevent the model from learning coarse topological structures, the initial features are often augmented with structural information of the graph, typically in the form of Laplacian eigenvectors or Random Walk transition probabilities. In this work, we explore the contribution of message passing when strong structural encodings are provided. We introduce a novel way of modeling the interaction between feature and structural information based on their tensor product rather than the standard concatenation. The choice of interaction is compared in common scenarios and in settings where the capacity of the message-passing layer is severely reduced and ultimately the message-passing phase is removed altogether. Our results indicate that using tensor-based encodings is always at least on par with the concatenation-based encoding and that it makes the model much more robust when the message passing layers are removed, on some tasks incurring almost no drop in performance. This suggests that the importance of message passing is limited when the model can construct strong structural encodings.

Bahrul Ilmi Nasution, Floor Eijkelboom, Mark Elliot et al.

Synthetic data generation is an important tool for privacy-preserving data sharing. While diffusion models have set recent benchmarks, flow matching (FM) offers a promising alternative. This paper presents different ways to implement flow matching for tabular data synthesis. We provide a comprehensive empirical study that compares flow matching (FM and variational FM) with a state-of-the-art diffusion method (TabDDPM and TabSyn) in tabular data synthesis. We evaluate both the standard Optimal Transport (OT) and the Variance Preserving (VP) probability paths, and also compare deterministic and stochastic samplers -- something possible when learning to generate using \textit{variational} flow matching -- characterising the empirical relationship between data utility and privacy risk. Our key findings reveal that flow matching, particularly TabbyFlow, outperforms diffusion baselines. Flow matching methods also achieves better performance with remarkably low function evaluations ($\leq$ 100 steps), offering a substantial computational advantage. The choice of probability path is also crucial, as using the OT path demonstrates superior performance, while VP has potential for producing synthetic data with lower disclosure risk. Lastly, our results show that making flows stochastic not only preserves marginal distributions but, in some instances, enables the generation of high utility synthetic data with reduced disclosure risk.

Jung Yeon Park, Yuxuan Chen, Floor Eijkelboom et al.

Symmetry is fundamental to understanding physical systems, and at the same time, can improve performance and sample efficiency in machine learning. Both pursuits require knowledge of the underlying symmetries in data. To address this, we propose learning symmetries directly from data via flow matching on Lie groups. We formulate symmetry discovery as learning a distribution over a larger hypothesis group, such that the learned distribution matches the symmetries observed in data. Relative to previous works, our method, \lieflow, is more flexible in terms of the types of groups it can discover and requires fewer assumptions. Experiments on 2D and 3D point clouds demonstrate the successful discovery of discrete groups, including reflections by flow matching over the complex domain. We identify a key challenge where the symmetric arrangement of the target modes causes ``last-minute convergence,'' where samples remain stationary until relatively late in the flow, and introduce a novel interpolation scheme for flow matching for symmetry discovery.

Daan Roos, Oscar Davis, Floor Eijkelboom et al.

We introduce Categorical Flow Maps, a flow-matching method for accelerated few-step generation of categorical data via self-distillation. Building on recent variational formulations of flow matching and the broader trend towards accelerated inference in diffusion and flow-based models, we define a flow map towards the simplex that transports probability mass toward a predicted endpoint, yielding a parametrisation that naturally constrains model predictions. Since our trajectories are continuous rather than discrete, Categorical Flow Maps can be trained with existing distillation techniques, as well as a new objective based on endpoint consistency. This continuous formulation also automatically unlocks test-time inference: we can directly reuse existing guidance and reweighting techniques in the categorical setting to steer sampling toward downstream objectives. Empirically, we achieve state-of-the-art few-step results on images, molecular graphs, and text, with strong performance even in single-step generation.

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.

Pedro M. P. Curvo, Maksim Zhdanov, Floor Eijkelboom et al.

Existing approaches to controllable generation typically rely on fine-tuning, auxiliary networks, or test-time search. We show that flow matching admits a different control interface: adaptation through examples. For deterministic interpolants, the velocity field is solely governed by a conditional endpoint mean; shifting this mean shifts the flow itself. This yields a simple principle for controllable generation: steer a pretrained model by changing the reference set it follows. We instantiate this idea in two forms. Reference-Mean Guidance is training-free: it computes a closed-form endpoint-mean correction from a reference bank and applies it to a frozen FLUX.2-klein (4B) model, enabling control of color, identity, style, and structure while keeping the prompt, seed, and weights fixed. Semi-Parametric Guidance amortizes the same idea through an explicit mean anchor and learned residual refiner, matching unconditional DiT-B/4 quality on AFHQv2 while allowing the reference set to be swapped at inference time. These results point to a broader direction: generative models that adapt through data, not parameter updates.

Cong Liu, David Ruhe, Floor Eijkelboom et al.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

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.

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.

Răzvan-Andrei Matişan, Vincent Tao Hu, Grigory Bartosh et al.

We introduce Purrception, a variational flow matching approach for vector-quantized image generation that provides explicit categorical supervision while maintaining continuous transport dynamics. Our method adapts Variational Flow Matching to vector-quantized latents by learning categorical posteriors over codebook indices while computing velocity fields in the continuous embedding space. This combines the geometric awareness of continuous methods with the discrete supervision of categorical approaches, enabling uncertainty quantification over plausible codes and temperature-controlled generation. We evaluate Purrception on ImageNet-1k 256x256 generation. Training converges faster than both continuous flow matching and discrete flow matching baselines while achieving competitive FID scores with state-of-the-art models. This demonstrates that Variational Flow Matching can effectively bridge continuous transport and discrete supervision for improved training efficiency in image generation.

Floor Eijkelboom, Grigory Bartosh, Christian Andersson Naesseth et al.

We present a formulation of flow matching as variational inference, which we refer to as variational flow matching (VFM). Based on this formulation we develop CatFlow, a flow matching method for categorical data. CatFlow is easy to implement, computationally efficient, and achieves strong results on graph generation tasks. In VFM, the objective is to approximate the posterior probability path, which is a distribution over possible end points of a trajectory. We show that VFM admits both the CatFlow objective and the original flow matching objective as special cases. We also relate VFM to score-based models, in which the dynamics are stochastic rather than deterministic, and derive a bound on the model likelihood based on a reweighted VFM objective. We evaluate CatFlow on one abstract graph generation task and two molecular generation tasks. In all cases, CatFlow exceeds or matches performance of the current state-of-the-art models.

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

Floor Eijkelboom, Rob Hesselink, Erik Bekkers

This paper presents $\mathrm{E}(n)$ Equivariant Message Passing Simplicial Networks (EMPSNs), a novel approach to learning on geometric graphs and point clouds that is equivariant to rotations, translations, and reflections. EMPSNs can learn high-dimensional simplex features in graphs (e.g. triangles), and use the increase of geometric information of higher-dimensional simplices in an $\mathrm{E}(n)$ equivariant fashion. EMPSNs simultaneously generalize $\mathrm{E}(n)$ Equivariant Graph Neural Networks to a topologically more elaborate counterpart and provide an approach for including geometric information in Message Passing Simplicial Networks. The results indicate that EMPSNs can leverage the benefits of both approaches, leading to a general increase in performance when compared to either method. Furthermore, the results suggest that incorporating geometric information serves as an effective measure against over-smoothing in message passing networks, especially when operating on high-dimensional simplicial structures. Last, we show that EMPSNs are on par with state-of-the-art approaches for learning on geometric graphs.