Revisiting Neural Processes via Fourier Transform and Volterra Series

2026-06-12 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Data Science & Analytics · Depth: Expert, extended

Summary

This research introduces Set Fourier Convolutions (SFConvs) and Set Fourier Volterra ConvCNPs (SFVConvCNPs) as novel approaches to address key limitations in existing translation-equivariant Neural Processes (NPs). Current methods either obscure function classes and interpretability with stacked non-linearities or suffer from quadratic scaling for attention-based models, or require dense uniform input grids for convolutional designs. The proposed framework leverages Volterra expansion to characterize continuous translation-equivariant operators, offering analytical transparency. SFConvs provide a frequency-domain parameterization that operates directly on irregularly sampled points, achieves approximately global receptive fields, and scales linearly with observations. Experiments across synthetic 1D regression, predator-prey systems, 2D image completion (CIFAR-10, SVHN), and 3D spatiotemporal tasks (Kolmogorov flow, ERA5 climate regression) demonstrate that SFConvCNPs and SFVConvCNPs consistently outperform or match state-of-the-art baselines, particularly excelling in generalization to geographically disjoint regions in climate data.

Key takeaway

For Machine Learning Engineers developing probabilistic functional models for irregularly sampled spatiotemporal data, you should evaluate Set Fourier Convolutional Neural Processes (SFConvCNPs). These models offer linear scalability and global receptive fields, outperforming traditional convolutional or attention-based Neural Processes in tasks like climate regression and image completion. Adopting SFConvCNPs can enhance model interpretability and generalization, especially where translation equivariance is critical.

Key insights

Combining Volterra series with frequency-domain convolutions yields scalable, interpretable translation-equivariant Neural Processes for irregular data.

Principles

• Translation equivariance improves sample efficiency and generalization.
• Volterra series characterize continuous translation-equivariant operators.
• Frequency-domain convolutions enable global receptive fields and linear scaling.

Method

Characterize continuous translation-equivariant operators via Volterra expansion. Introduce Set Fourier Convolutions (SFConvs) for frequency-domain parameterization on irregular data, then build SFConvCNPs and SFVConvCNPs.

In practice

• Model complex dynamical systems from irregular measurements.
• Improve climate prediction with better spatial generalization.
• Enhance image completion on 2D datasets.

Topics

• Neural Processes
• Translation Equivariance
• Fourier Transform
• Volterra Series
• Spatiotemporal Data
• Probabilistic Modeling

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.