Grounding Continuous Representations in Geometry: Equivariant Neural Fields (2406.05753v5)

Abstract: 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.

• The paper introduces Equivariant Neural Fields, integrating geometric conditioning into Neural Fields for improved interpretability and performance.
• It employs latent point clouds and cross-attention transformers to enforce bi-invariance and steerability, ensuring consistent geometric transformations.
• Empirical results demonstrate enhanced reconstruction quality, higher classification accuracy, and novel local editing capabilities across multiple datasets.

Grounding Continuous Representations in Geometry: Equivariant Neural Fields

The paper "Grounding Continuous Representations in Geometry: Equivariant Neural Fields" proposes a novel approach to enhancing the performance of Neural Fields (NeFs) by introducing Equivariant Neural Fields (ENFs). The primary objective of this work is to address the limitations of traditional NeFs, particularly their lack of geometric interpretability. The authors achieve this by grounding NeFs in geometric point clouds and leveraging cross-attention transformers, resulting in improved performance and new capabilities for geometric reasoning and local field editing.

Key Contributions and Methodology

The central innovation of the paper is the incorporation of geometric variables into the conditioning of NeFs. Specifically, the authors propose using latent point clouds, composed of poses and context vectors, to condition the continuous representations. This model induces a steerability property, ensuring that transformations in the field correlate with transformations in the latent space.

The paper makes several significant contributions:

1. Equivariance and Steerability: The proposed model enforces a bi-invariance constraint, ensuring that fields and their latent representations transform consistently. This is essential for maintaining geometric coherence across transformations.
2. Localized Representations: By structuring the latent representations as localized point clouds, the model can efficiently share weights over similar local patterns, enhancing learning efficiency and interpretability.
3. Empirical Validation: The authors validate their approach through extensive experiments, demonstrating improved classification performance, better reconstruction quality, and unique local field editing capabilities.

Theoretical Underpinnings

The steerability property is formally defined as: $$\forall g \in G: f_\theta(g^{-1}x, z) = f_\theta(x, gz)$$ This property requires that the neural field $f_\theta$ is bi-invariant with respect to both the coordinates $x$ and the pose $p$. The authors leverage group theory, particularly the Special Euclidean group $SE(n)$, to formalize and enforce this property. The conditioning variables are modeled as geometric point sets, ensuring that the representations adhere to the same group transformation laws as the fields.

Implementation and Results

The ENFs are implemented using cross-attention mechanisms, where the attention scores are modulated by geometric invariants derived from the latent poses. The authors introduce a Gaussian window to localize attention around the poses, ensuring that the latent point sets represent localized regions in the input space.

Experimental results across multiple datasets (Cifar10, CelebA, STL-10, and ShapeNet) showcase the superiority of ENFs over traditional NeFs and the functa framework:

• Reconstruction Performance: ENFs demonstrate higher Peak Signal-to-Noise Ratio (PSNR) values compared to baseline methods, indicating better reconstruction quality.
• Classification Accuracy: The grounded representations enable more effective downstream classification tasks, with higher accuracy achieved using ENFs.
• Local Field Editing: The structured latent representations facilitate unique local editing capabilities, such as stitching or merging parts of different fields, which are not possible with traditional NeFs.

Implications and Future Directions

The proposed ENF framework has significant implications for various applications involving continuous signal representations, such as computer vision, geometric deep learning, and spatio-temporal data modeling. By grounding representations in geometry and ensuring equivariance, ENFs provide a robust toolset for analyzing and manipulating geometric data.

Theoretical implications include a deeper understanding of how geometric inductive biases can be integrated into neural network architectures to improve learning efficiency and interpretability. Practically, the enhanced ability to perform geometric reasoning opens new avenues for advanced tasks like dynamic field modeling and shape analysis.

Future research could explore extending the ENF framework to other types of geometric structures and investigating its applicability to broader datasets and domains. Additionally, optimizing the computational efficiency and scalability of ENFs for large-scale applications remains an important direction.

In conclusion, this paper presents a substantial advancement in the representation of continuous signals by integrating geometric grounding and equivariance principles, demonstrating both theoretical and practical benefits in diverse machine learning tasks.