Revisiting Neural Processes via Fourier Transform and Volterra Series
Authors: Peiman Mohseni, Nick Duffield, Raymond K. W. Wong
Organizations: Texas A&M University, College Station, Texas, USA.
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 are limited by local receptive fields and the need to embed inputs onto a dense uniform grid, while attention-based alternatives lift these restrictions at quadratic cost in the number of observations. We address both with two contributions. First, using the Volterra expansion, we approximate continuous translation-equivariant operators by sums of higher-order convolutions, yielding analytical transparency while admitting efficient evaluation via 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.
What functions can Neural Processes represent? We analyze the representational capacity of popular NP architectures: Conditional Neural Processes (CNPs), Attentive Neural Processes (ANPs), Transformer Neural Processes (TNPs), and their latent variants. We prove these architectures form a strict hierarchy. CNP-representable functions are exactly those depending on finitely many expected features of the context distribution. ANPs strictly generalize CNPs via query-dependent reweighting, enabling kernel smoothers. ConvCNPs and ANPs are incomparable; each contains functions outside the other, separated by stationarity versus translation equivariance. TNPs with L self-attention layers capture L-hop context interactions. For latent NPs, we show finite-dimensional latents provide coherent sampling but do not circumvent encoder limitations; matching GP posterior distributions requires latent dimension scaling with context size. These results provide a theoretical foundation for architecture selection based on task structure.
Time series, spatial data, and images are natural applications of Neural Processes. However, when such data exhibit strong periodicity and quasi-periodicity, existing methods often suffer from underfitting and generalise poorly beyond the training distribution. In this work, we propose Spectral Transformer Neural Processes (STNPs), a frequency-aware extension of Transformer Neural Processes (TNPs). STNPs introduce a Spectral Aggregator that estimates an empirical context spectrum, compresses it into a spectral mixture, samples task-adaptive spectral features, and concatenates them with time-domain embeddings, thereby injecting a spectral-mixture-kernel bias into TNPs. This design reshapes the similarity geometry, allowing inputs that are distant in Euclidean space to remain close in an induced periodic manifold while enhancing time-frequency interactions. Extensive experiments on synthetic regression tasks, real-world time-series datasets, and an image dataset demonstrate that STNPs consistently improve predictive performance over existing baselines, extending Neural Processes beyond translation equivariance towards effective modelling of periodicity and quasi-periodicity.
Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within the model are Fourier layers, which apply linear transformations directly to the Fourier modes, with parameters depending on the wave numbers. However, most physical systems are isotropic, with the results being independent of the coordinate system chosen, but the linear transformations do not necessarily respect these symmetries. We propose a modification to the linear transformations that ensures spatial symmetries are respected, called the Isotropic Fourier Neural Operator, which both improves model performance and reduces the number of parameters by up to a factor of 16 in 2D and 96 in 3D.