Grounding Continuous Representations in Geometry: Equivariant Neural Fields
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| Format: | Preprint |
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2024
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| _version_ | 1866929701540331520 |
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| author | Wessels, David R Knigge, David M Papa, Samuele Valperga, Riccardo Vadgama, Sharvaree Gavves, Efstratios Bekkers, Erik J |
| author_facet | Wessels, David R Knigge, David M Papa, Samuele Valperga, Riccardo Vadgama, Sharvaree Gavves, Efstratios Bekkers, Erik J |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_05753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grounding Continuous Representations in Geometry: Equivariant Neural Fields Wessels, David R Knigge, David M Papa, Samuele Valperga, Riccardo Vadgama, Sharvaree Gavves, Efstratios Bekkers, Erik J Machine Learning Artificial Intelligence Computer Vision and Pattern Recognition 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. |
| title | Grounding Continuous Representations in Geometry: Equivariant Neural Fields |
| topic | Machine Learning Artificial Intelligence Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2406.05753 |