Graph Variate Neural Networks

Abstract

Modelling dynamically evolving spatio-temporal signals is a prominent challenge in the Graph Neural Network (GNN) literature. Notably, GNNs assume an existing underlying graph structure. While this underlying structure may not always exist or is derived independently from the signal, a temporally evolving functional network can always be constructed from multi-channel data. Graph Variate Signal Analysis (GVSA) defines a unified framework consisting of a network tensor of instantaneous connectivity profiles against a stable support usually constructed from the signal itself. Building on GVSA and tools from graph signal processing, we introduce Graph-Variate Neural Networks (GVNNs): layers that convolve spatio-temporal signals with a signal-dependent connectivity tensor combining a stable long-term support with instantaneous, data-driven interactions. This design captures dynamic statistical interdependencies at each time step without ad hoc sliding windows and admits an efficient implementation with linear complexity in sequence length. Across forecasting benchmarks, GVNNs consistently outperform strong graph-based baselines and are competitive with widely used sequence models such as LSTMs and Transformers. On EEG motor-imagery classification, GVNNs achieve strong accuracy highlighting their potential for brain-computer interface applications.

Figures (8)

• Figure 1: Graph Variate Fourier Transform (GVFT). Each panel shows the GVFT coefficients of a synthetic multivariate time series projected onto the eigenbasis of its own graph‐structured connectivity profile at each time step. The left heatmap uses a squared‐difference formulation for $\Omega_{t} = (x_{i} - x_{j})^{2} \cdot C$, while the right uses instantaneous correlation: $\Omega_{t} = \mathrm{corr}(x_{t}) \cdot C$, where $C$ is the long‐term correlation matrix across the full signal. The GVFT transforms the input signal $X \in \mathbb{R}^{N \times T}$ into a new matrix $\hat{X} \in \mathbb{R}^{N \times T}$, where each column represents the projection of $x_{t}$ onto the eigenbasis of $\Omega_{t}$. This figure illustrates how different formulations of signal‐derived connectivity affect the spectral content and dynamics of the transformed signal.
• Figure 2: Graph-Variate Neural Network (GVNN) layer. A multivariate sequence $X\in\mathbb{R}^{N\times T}$ induces instantaneous connectivity $J(t)$, which is combined with a long-term support $W$ to form $\Omega(t)=W\circ J(t)$. In parallel, $X$ and $\Omega(t)$ drive a batched graph convolution $Z(t)=\Omega(t)X(t)$. A skip connection carries $X$ to the combiner, which applies a learned linear map and a nonlinearity, $Y(t)=\sigma\!(\Theta[\,a_t X(t)+b_t Z(t)\,])$.
• Figure 3: Comparison of instantaneous correlation profile, long-term covariance, and Hadamard-filtered covariance matrices. Each panel displays the respective matrix with its condition number and invertibility status.
• Figure 4: Instantaneous Connectivity Profiles (Inst. Correlation). Transient Brain states are not stationary. This figure shows how, depending on the temporal position in the time-locked task, connectivity can change significantly. GVNNs exploit this, allowing a framework for the analysis of Dynamic Functional Connectivity.
• Figure 5: Learned graph support matrix $W_C$ before and after training. The figure illustrates how the static graph support matrix $W_C$ evolves through training. The left panel shows the initialized matrix, while the right panel presents the learned weights after optimization, revealing how the model adapts graph connectivity structure for improved forecasting.

Theorems & Definitions (29)

• Definition 1: Graph Convolutional Filter
• Definition 2: Graph Fourier Transform (GFT)
• Definition 3: Graph Convolutional Network (GCN)
• Definition 4: Graph-Time Convolutional Neural Network (GTCNN) Isufi2021GTCNNSabbaqi2023GTCNN
• Definition 5: Graph-Variate Signal Analysis
• Definition 6: Graph-Variate Neural Network (GVNN, layer-wise form)
• Definition 7: Graph Variate Fourier Transform
• Definition 8: Graph-Variate frequency response
• Theorem 1: Parseval identity for the GVFT
• proof