Author name cluster

David M. Knigge

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4 papers

2 author rows

ICLR Conference 2025 Conference Paper

Grounding Continuous Representations in Geometry: Equivariant Neural Fields

David R. Wessels
David M. Knigge
Riccardo Valperga
Samuele S. Papa
Sharvaree Vadgama
Efstratios Gavves
Erik J. Bekkers

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 faitfhully, allowing for geometric reasoning in latent space, (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.

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NeurIPS Conference 2024 Conference Paper

Space-Time Continuous PDE Forecasting using Equivariant Neural Fields

David M. Knigge
David R. Wessels
Riccardo Valperga
Samuele Papa
Jan-Jakob Sonke
Efstratios Gavves
Erik J. Bekkers

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 -- such as 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 of the Conditional NeF - preserves 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. Furthermore, we validate 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.

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ICLR Conference 2023 Conference Paper

Modelling Long Range Dependencies in $N$D: From Task-Specific to a General Purpose CNN

David M. Knigge
David W. Romero
Albert Gu
Efstratios Gavves
Erik J. Bekkers
Jakub M. Tomczak
Mark Hoogendoorn
Jan-Jakob Sonke

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.

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ICML Conference 2022 Conference Paper

Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups

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 of the plane. $\mathrm{Sim(2)}$-equivariance further improves performance on all tasks considered, and achieves state-of-the-art performance on rotated MNIST.

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