[Paper Review] Generalised Implicit Neural Representations

Daniele Grattarola, Pierre Vandergheynst|arXiv (Cornell University)|May 31, 2022

Neural Networks and Applications4 citations

TL;DR

This paper proposes Generalised Implicit Neural Representations (GINRs) to model signals on non-Euclidean domains without relying on coordinate systems. By using spectral embeddings from the graph Laplacian as coordinate-free input to a neural network, GINRs learn continuous signal representations that generalize across different graph realizations and enable super-resolution and conditional modeling, demonstrated on biological, social, and meteorological data with high-fidelity reconstructions and R² > 0.9 on protein electrostatic signals.

ABSTRACT

We consider the problem of learning implicit neural representations (INRs) for signals on non-Euclidean domains. In the Euclidean case, INRs are trained on a discrete sampling of a signal over a regular lattice. Here, we assume that the continuous signal exists on some unknown topological space from which we sample a discrete graph. In the absence of a coordinate system to identify the sampled nodes, we propose approximating their location with a spectral embedding of the graph. This allows us to train INRs without knowing the underlying continuous domain, which is the case for most graph signals in nature, while also making the INRs independent of any choice of coordinate system. We show experiments with our method on various real-world signals on non-Euclidean domains.

Motivation & Objective

• To address the limitation of existing implicit neural representations (INRs) that rely on Euclidean coordinates and fail on non-Euclidean domains such as graphs and manifolds.
• To develop a method that learns signal representations independent of any coordinate system, enabling generalization across different graph realizations of the same underlying domain.
• To enable high-resolution signal reconstruction and conditional modeling on irregular, non-Euclidean domains using a single, learnable neural network.

Proposed method

• Approximate the location of sampled nodes on an unknown topological space using spectral embeddings derived from the eigenvectors of the graph Laplacian.
• Train a neural network to map each node’s spectral embedding to the corresponding signal value, effectively learning an implicit representation of the signal.
• Use the spectral embedding as a coordinate-free input, making the INR invariant to arbitrary coordinate system choices.
• Leverage the fact that graph Laplacian eigenvectors approximate continuous eigenfunctions of the Laplace-Beltrami operator, enabling generalization to new points on the same domain.
• Apply Fourier features or positional encodings implicitly through the spectral embedding, generalizing the use of positional encodings in Euclidean INRs.
• Train conditional GINRs by conditioning the network on a latent vector to represent multiple signals on the same or different domains.

Experimental results

Research questions

• RQ1 Can implicit neural representations be generalized to signals on non-Euclidean domains such as graphs and manifolds without relying on a coordinate system?
• RQ2 How can spectral embeddings from the graph Laplacian serve as a coordinate-free alternative to Euclidean coordinates in INR training?
• RQ3 To what extent can a single GINR model generalize across different graph realizations of the same underlying signal?
• RQ4 Can GINRs enable super-resolution and accurate reconstruction of signals from low-resolution graph samples?
• RQ5 Can conditional GINRs model diverse signals on the same or different irregular domains using a shared architecture?

Key findings

• The GINR model achieved R² ≥ 0.9 on predicting electrostatic signals across 500 different protein surfaces using a single set of network weights and a latent vector of size 8.
• The model successfully super-resolved meteorological signals on a spherical Earth domain, producing high-resolution predictions at 30-minute temporal intervals from 1-hour training data.
• GINRs trained on one graph realisation could generalize to other graph realisations of the same underlying domain, demonstrating robustness to structural variation.
• The method enabled conditional modeling of multiple signals on the same irregular domain, including temperature, cloud cover, and wind speed on Earth’s surface.
• The spectral embedding-based approach allowed for continuous, differentiable signal approximation across arbitrary points on the domain, even when the original domain was unknown or irregular.
• The animated high-resolution output of the meteorological model confirmed the realism and temporal consistency of the learned signal dynamics.