# Inverse Cross-spectral Neural Networks for Multivariate Time Series

arXiv: https://arxiv.org/abs/2610.06630 · HTML: https://mlsanity.com/paper/2610.06630/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 0 of its 19 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Abstract (arXiv)

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.