Revisiting Neural Processes via Fourier Transform and Volterra Series

• URL: http://arxiv.org/abs/2606.01172v1
• Date: Sun, 31 May 2026 11:27:48 GMT
• Title: Revisiting Neural Processes via Fourier Transform and Volterra Series
• Abstract summary: Neural processes (NPs) are a family of probabilistic functional models.<n>NPs face two limitations: (i) they stack generic components with non-linearities, obscuring the induced function class and limiting interpretability.<n>We propose two conditional NPs (CNPs): SFConvCNPs, which stack SFConv blocks with non-linearities, and SFVConvCNPs, which integrate the Volterra formulation.
• Score: 8.061821280531884
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains [56.121725064621295]
  Score-based diffusion models in function spaces offer a principled framework for modelling function-valued data.<n>However, practical implementations have struggled to fully realise these benefits.<n>We propose a novel architecture that represents generalised convolutional kernels as finite graph kernels.
  arXiv  Detail & Related papers  (2026-05-05T08:33:52Z)
• A Unified Poisson Summation Framework for Generalized Quantum Matrix Transformations [2.1496959255800188]
  We present a unified algorithmic framework for quantum simulation of non-unitary dynamics and matrix functions.<n>By reinterpreting discretization errors as spectral folding in dual domains, we synthesize two distinct algorithmic paths.<n>We demonstrate the versatility of this framework by efficiently simulating diverse phenomena.
  arXiv  Detail & Related papers  (2026-04-03T08:42:08Z)
• Variational Entropic Optimal Transport [67.76725267984578]
  We propose Variational Entropic Optimal Transport (VarEOT) for domain translation problems.<n>VarEOT is based on an exact variational reformulation of the log-partition $log mathbbE[exp(cdot)$ as a tractable generalization over an auxiliary positive normalizer.<n> Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality.
  arXiv  Detail & Related papers  (2026-02-02T15:48:44Z)
• LFA applied to CNNs: Efficient Singular Value Decomposition of Convolutional Mappings by Local Fourier Analysis [4.69726714177332]
  singular values of convolutional mappings encode interesting spectral properties.<n> computation of singular values is typically very resource-intensive.<n>We propose an approach of complexity O(N) based on local Fourier analysis.
  arXiv  Detail & Related papers  (2025-06-05T22:10:01Z)
• Spectral Convolutional Conditional Neural Processes [4.52069311861025]
  Conditional Neural Processes (CNPs) constitute a family of probabilistic models that harness the flexibility of neural networks to parameterize processes. We propose Spectral Convolutional Conditional Neural Processes (SConvCNPs), a new addition to the NPs family that allows for more efficient representation of functions in the frequency domain.
  arXiv  Detail & Related papers  (2024-04-19T21:13:18Z)
• D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory [79.50644650795012]
  We propose a deep learning approach to solve Kohn-Sham Density Functional Theory (KS-DFT) We prove that such an approach has the same expressivity as the SCF method, yet reduces the computational complexity. In addition, we show that our approach enables us to explore more complex neural-based wave functions.
  arXiv  Detail & Related papers  (2023-03-01T10:38:10Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling.<n>This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space.<n>We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Contrastive Conditional Neural Processes [45.70735205041254]
  Conditional Neural Processes(CNPs) bridge neural networks with probabilistic inference to approximate functions of Processes under meta-learning settings. Two auxiliary contrastive branches are set up hierarchically, namely in-instantiation temporal contrastive learning(tt TCL) and cross-instantiation function contrastive learning(tt FCL) We empirically show that tt TCL captures high-level abstraction of observations, whereas tt FCL helps identify underlying functions, which in turn provides more efficient representations.
  arXiv  Detail & Related papers  (2022-03-08T10:08:45Z)
• Efficient Semi-Implicit Variational Inference [65.07058307271329]
  We propose an efficient and scalable semi-implicit extrapolational (SIVI) Our method maps SIVI's evidence to a rigorous inference of lower gradient values.
  arXiv  Detail & Related papers  (2021-01-15T11:39:09Z)
• Central and Non-central Limit Theorems arising from the Scattering Transform and its Neural Activation Generalization [4.125187280299247]
  We study a generalization of the scattering transform (ST) that includes broad neural activation functions, which is called neural activation ST (NAST) Here, the nonlinear function models how a neuron gets excited by the input signal. Results explain how NAST processes complicated and non-stationary time series, and pave a way towards statistical inference based on NAST under the non-null case.
  arXiv  Detail & Related papers  (2020-11-21T14:31:57Z)
• Probabilistic Numeric Convolutional Neural Networks [80.42120128330411]
  Continuous input signals like images and time series that are irregularly sampled or have missing values are challenging for existing deep learning methods. We propose Probabilistic Convolutional Neural Networks which represent features as Gaussian processes (GPs) We then define a convolutional layer as the evolution of a PDE defined on this GP, followed by a nonlinearity. In experiments we show that our approach yields a $3times$ reduction of error from the previous state of the art on the SuperPixel-MNIST dataset and competitive performance on the medical time2012 dataset PhysioNet.
  arXiv  Detail & Related papers  (2020-10-21T10:08:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.