CoVariance Neural Networks and their extensions have emerged as effective tools for processing multivariate data, deriving graph shift operators directly from second-order statistics. These architectures, however, are designed for independent and identically distributed observations and do not fully capture the joint structure of temporal and cross-variable dependencies in multivariate time series. In this work, we introduce Inverse Cross-Spectral Neural Networks (iCSNNs), a class of graph neural networks for stationary multivariate time series whose shift operators are the inverse cross-spectral density (iCSD) matrices. These operators encode frequency-specific conditional relationships among variables, exploiting the decomposition provided by the spectral representation theorem. Leveraging spectral smoothness, frequencies are grouped into bands sharing a single iCSD operator, yielding a compact parametrisation that retains the frequency-dependent structure of the process. We further propose a joint learning procedure to estimate both the Fourier-domain dependence structure and the iCSNN parameters, adapting the iCSD operators to the downstream task. When tested on synthetic data, iCSNN outperforms baselines from different methodological families.
Figures & tables
Figure 1 : The pipeline of the iCSNN.
Figure 2 : Macro-averaged F1 scores of the jointly trained iCSNN, its Disjoint ablation and baseline methods, averaged over 20 synthetic datasets. The red dashed line represents the chance-level score of 0.20 for five balanced classes.
Multivariate time series contain complex patterns that span across both space and time. While covariance-based statistical tools like spatiotemporal Principal Component Analysis (ST-PCA) help identify these patterns, they are limited to linear operations and prone to estimation errors with limited data. Recent covariance-based spatiotemporal neural networks offer more stable, non-linear alternatives, but they ignore correlations across different time steps. To solve this, we introduce the Kronecker coVariance Neural Network (KVNN), a temporal graph neural network that represents the spatiotemporal covariance matrix via a sum of Kronecker products where spatial and temporal dependencies are decoupled. By implementing filtering operations on spatial and temporal components, KVNNs achieve expressive processing capabilities, admit a rigorous spectral analysis, and are provably stable to finite-sample estimation errors, ultimately addressing all of ST-PCA's limitations. We show on five real-world datasets that KVNNs achieve strong forecasting performance, often requiring significantly fewer trainable parameters than competitive methods, and are consistent under estimation noise.
Andrea Cavallo, Athanasios Georgoutsos, Elvin Isufi
Delft University of Technology, Delft, Netherlands
Spiking Neural Networks (SNNs) have emerged as an energy-efficient alternative to conventional neural networks, demonstrating strong performance in computer vision and robotics. More recently, SNNs have been applied to time series forecasting (TSF), with methods exploring spiking temporal backbones, spike-compatible positional encodings, Fourier-domain processing, and redesigned neuron dynamics. However, existing SNN forecasting approaches process variables independently, lacking explicit mechanisms for modeling inter-variable dependencies. This is a critical limitation in multivariate settings, where cross-variable correlations carry substantial predictive information. We propose Spiking Fourier Graph Operators (SpikF-GO), which addresses this gap by combining a hypervariate graph formulation in which every scalar observation becomes a graph node with spike-driven spectral processing. SpikF-GO introduces a Hard Concrete frequency gate for learnable sparse frequency selection and a Complex LIF gate that applies independent spiking neurons to real and imaginary Fourier components, preserving binary, event-driven computation throughout the spectral domain. We further present a variant incorporating Central Pattern Generator-based positional encodings for stronger long-range temporal modeling. Evaluated on eight benchmarks under a unified experimental protocol, SpikF-GO achieves the best average rank among all SNN methods and outperforms its ANN counterpart, FourierGNN, at reduced energy cost. SpikF-GO maintains competitive accuracy even at substantially smaller embedding dimensions, thereby achieving significant energy reductions. To our knowledge, this is among the first works to bring graph-based multivariate modeling into the spiking domain for TSF and the first to provide a unified comparison across SNN forecasting architectures under a common experimental protocol.
Jafar Bakhshaliyev, Niels Landwehr
Data Science Group, University of Hildesheim, Hildesheim, Germany
Multivariate time series forecasting presents unique challenges because future variables often co-evolve under shared system dynamics. While existing studies mainly focus on cross-variable dependencies in historical observations, dependencies among future values are much less explored. Specifically, modern forecasting models largely follow the Direct Forecasting (DF) paradigm, generating multi-step forecasts with point-wise objectives that do not explicitly constrain cross-variable structure. In this work, we show that the DF objective is mismatched in the presence of cross-variable and lagged dependencies, revealing an objective gap. To address this issue, we propose \textbf{C}ross-\textbf{V}ariable \textbf{Loss} (CvLoss), a plug-in structural regularizer that constrains forecast residuals on a cross-variable graph. CvLoss penalizes inconsistent edge-wise residual differences over forecast patches, encouraging consistency across both synchronous and asynchronous interactions. Our experiments show that CvLoss consistently improves competitive forecasting models, outperforms representative learning objectives, and is compatible with a variety of forecasting backbones.
Kuiye Ding, Yifan Hu, Hanchen Wang +1
University of Technology Sydney · Tsinghua University · The Hong Kong University of Science and Technology (Guangzhou)