Revisiting Neural Processes via Fourier Transform and Volterra Series
This work provides a more interpretable and efficient framework for translation-equivariant neural processes, benefiting applications in scientific and engineering domains requiring modeling of irregularly sampled functions.
The authors address limitations in translation-equivariant neural processes by characterizing continuous translation-equivariant operators via Volterra expansion and introducing set Fourier convolutions (SFConvs) that operate on irregularly sampled points with linear scaling. Their proposed models (SFConvCNPs and SFVConvCNPs) achieve competitive performance against state-of-the-art baselines on synthetic and real-world datasets.
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.