Revisiting Neural Processes via Fourier Transform and Volterra Series

2026-05-31

AI Summary

This work addresses the limitations of existing translation-equivariant neural processes, which often suffer from poor interpretability, reliance on dense grids, or high computational complexity when modeling finite and irregularly sampled observations. To overcome these issues, the paper introduces a continuous translation-equivariant operator modeling paradigm based on Volterra series and, for the first time, incorporates it into the functional class analysis of neural processes. By designing a set-based Fourier convolution (SFConv) in the frequency domain, the proposed method directly handles irregular sampling points while maintaining linear computational scalability and achieving an approximately global receptive field. Experimental results demonstrate that the approach significantly outperforms current state-of-the-art baselines on both synthetic and real-world datasets, exhibiting superior sample efficiency, generalization capability, and scalability.

Abstract

Modeling unknown latent functions from finite, irregularly sampled measurements is a recurring challenge across science and engineering. Neural processes (NPs), a family of probabilistic functional models, are promising solutions -- especially when endowed with domain-specific symmetries like translation equivariance, which improve sample efficiency and generalization. Yet existing translation-equivariant NPs face two limitations: (i) they stack generic components with non-linearities, obscuring the induced function class and limiting interpretability; and (ii) convolutional designs rely on kernels with local receptive fields and require dense uniform input grids, while attention-based methods avoid these issues but scale quadratically with the number of observations. We address both with two contributions. First, using the Volterra expansion, we characterize continuous translation-equivariant operators as sums of higher-order convolutions, yielding analytical transparency while admitting efficient approximation by first-order convolutions. Second, we introduce set Fourier convolutions (SFConvs), a frequency-domain parameterization that operates directly on irregularly sampled points, achieves approximately global receptive fields, and scales linearly in the number of observations. Building on these ideas, we propose two conditional NPs (CNPs): SFConvCNPs, which stack SFConv blocks with non-linearities, and SFVConvCNPs, which integrate the Volterra formulation. Experiments on synthetic and real-world datasets demonstrate our methods' efficacy against state-of-the-art baselines.

Problem

• neural processes
• translation equivariance
• irregular sampling
• function modeling
• computational scalability

Innovation

• Volterra series
• Fourier transform
• translation equivariance
• set convolution
• neural processes