Research map / Probabilistic and scientific modeling / Variational and generative sampling
Grounding Continuous Representations in Geometry: Equivariant Neural Fields
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes Equivariant Neural Fields (ENFs), a conditional neural field whose latent is a geometric point cloud of pose-appearance tuples, decoded via bi-invariant geometry-informed cross-attention to give steerability and weight sharing.
Problem
Conditional neural fields compress each signal into a single global latent, which lacks explicit geometric structure and locality, limiting performance on tasks needing fine-grained geometric reasoning such as classification, segmentation, and forecasting.
Approach
Condition the neural field on a latent point cloud of (pose, context) tuples through cross-attention whose queries, keys and values depend only on bi-invariant attributes of coordinate-pose pairs, plus a Gaussian spatial window and k-NN restriction, so that transforming the field transforms the latent accordingly (steerability).
Key results
- On CIFAR10, ENF with translation-equivariant attributes reaches 82.1% classification accuracy versus 68.3% for Functa, and 42.2 dB reconstruction PSNR versus 38.1 dB.
- On ShapeNet16 voxel reconstruction, ENF a^R3 achieves 92.9 IoU and 96.6% classification accuracy, compared with 92.1 and 90.3 for Functa.
- On ImageNet1K reconstruction, ENF variants reach 24.7-27.5 dB PSNR versus 7.2 dB for Functa.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use translationally equivariant ENF instead of Functa | CIFAR10 representations · CIFAR10 test | accuracy: 68.3 -> 82.1 | Helped |
| use translationally equivariant ENF instead of Functa | CIFAR10 reconstruction · CIFAR10 test | PSNR: 38.1 -> 42.2 | Helped |
| use translationally equivariant ENF instead of Functa | CelebA 64×64 reconstruction · CelebA 64×64 test | PSNR: 28.0 -> 34.6 | Helped |
| use translationally equivariant ENF instead of Functa | ImageNet1K reconstruction · ImageNet1K test | PSNR: 7.2 -> 27.5 | Helped |
| use translationally equivariant ENF instead of Functa | ShapeNet55 reconstruction · ShapeNet55 test | IoU: 25.7 -> 73.2 | Helped |
| remove equivariance constraints from the ENF (ablation) | CIFAR10 reconstruction and classification · CIFAR10 test | accuracy: 82.1 -> 68.7 | Hurt |
| use roto-translational instead of translational equivariance (ablation) | CIFAR10 ENF · CIFAR10 test | accuracy: 82.1 -> 81.5 | Hurt |
| use translational equivariance instead of no-equivariance attributes (ablation) | CelebA reconstruction · CelebA 64×64 test | PSNR: 30.6 -> 34.6 | Helped |
| use translational equivariance instead of no-equivariance attributes (ablation) | ImageNet1K reconstruction · ImageNet1K test | PSNR: 24.7 -> 27.5 | Helped |
| use translational equivariance instead of no-equivariance attributes (ablation) | ShapeNet16 classification · ShapeNet16 test | accuracy: 96.4 -> 96.6 | No clear effect |
| increase latent count while reducing latent dimension at roughly fixed parameter count (ablation) | CIFAR10 ENF · CIFAR10 | accuracy: 53.21 -> 64.98 -> 73.54 -> 77.09 | Helped |
| add kNN approximation to attention (ablation) | CIFAR10 ENF without Gaussian spatial windowing · CIFAR10 | accuracy: 70.1 -> 72.8 | Helped |
Surprising findings
- A geometry-free ENF variant with pose-free latents reaches only 47.9% CIFAR10 accuracy and 22.3 dB PSNR, suggesting geometry in the latent is essential.
- ShapeNet-Part segmentation results are comparable between ENF and Functa, which the authors attribute to global alignment of shapes; a model conditioned only on object class reaches 64.3 class mIoU.
Limitations
- Computational cost scales with number of latents times number of coordinates, requiring k-NN approximation.
- Generative modelling via DiT on ENF latents underperforms dedicated generative field frameworks.
- Functa baseline and ENF struggle to reconstruct SDF shapes; meta-learning was insufficient for shape data, requiring autodecoding.
Experimental setup
Datasets: CIFAR10 CelebA 64x64 ImageNet1K ShapeNet-Part (ShapeNet16) ShapeNetCore v2 OMBRIA ERA5
Benchmarks: CIFAR10 CelebA 64x64 ImageNet1K ShapeNet16 ShapeNetCore v2 ShapeNet-Part OMBRIA ERA5
Metrics: PSNR accuracy IoU FID
Baselines: Functa NF2vec multi-modal U-Net PointNet-style point cloud architectures
Builds on: Functa PΘNITA Conditional Neural Fields Gaussian windowed attention DiT
Closest papers by meaning
- Connecting Neural Models Latent Geometries with Relative Geodesic Representations (NeurIPS 2025 · similarity 0.88)
- Neural Isometries: Taming Transformations for Equivariant ML (NeurIPS 2024 · similarity 0.87)
- How to Train Neural Field Representations: A Comprehensive Study and Benchmark (CVPR 2024 · similarity 0.86)
- Latent Functional Maps: a spectral framework for representation alignment (NeurIPS 2024 · similarity 0.86)
- CoFie: Learning Compact Neural Surface Representations with Coordinate Fields (NeurIPS 2024 · similarity 0.86)
- Space-Time Continuous PDE Forecasting using Equivariant Neural Fields (NeurIPS 2024 · similarity 0.86)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.