Grounding Continuous Representations in Geometry: Equivariant Neural Fields
This work addresses a bottleneck in geometric reasoning for researchers and practitioners using neural fields, though it appears incremental as it builds on existing CNF architectures.
The paper tackled the problem of Conditional Neural Fields (CNFs) lacking explicit geometric modeling in latent spaces, which limits fine-grained reasoning tasks like classification and segmentation, and proposed Equivariant Neural Fields (ENFs) to ground representations in geometry, resulting in clear improvements over baselines across multiple tasks.
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.