[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84503-en":3,"doc-seo-84503-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84503,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Revisiting Neural Processes via Fourier Transform and Volterra Series","Modeling unknown latent functions from finite, irregularly sampled measurements is a recurring problem in science and engineering. Neural processes (NPs) offer probabilistic function modeling and improve sample efficiency with translation equivariance, but existing approaches limit interpretability and impose scalability bottlenecks from either local convolutional receptive fields or quadratic-cost attention. The work introduces Volterra-series approximations for clearer operator structure and proposes set Fourier convolutions for frequency-domain, irregular-point modeling with linear scaling. Experiments on synthetic and real datasets validate performance against strong baselines.","Revisiting Neural Processes via Fourier Transform and Volterra Series  \nPeiman Mohseni 1 Nick Duffield 1 Raymond K. W. Wong 1  \narXiv :2606 .0 1 172v 3 [ cs .LG] 13 Jul 2026  \nAbstract  \nModeling 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 translationequivariant 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 realworld datasets demonstrate our methods’ efficacy against state-of-the-art baselines.  \n1Texas A&M University, College Station, Texas, USA. Correspondence to: Peiman Mohseni \u003C[peiman.mohseni@tamu.edu](peiman.mohseni@tamu.edu) >. Code available at [https://github.com/peiman-m/](https://github.com/peiman-m/)[ ](https://github.com/peiman-m/)[fourier-volterra-nps](fourier-volterra-nps.)[.](fourier-volterra-nps.)  \nProceedings of the 43 rd International Conference on Machine Learning, Seoul, South Korea. PMLR 306, 2026 . Copyright 2026 by the author(s) .  \n1. Introduction  \nMany scientific and engineering domains—such as turbulence modeling and climate science (Ravuri et al., 2021 ; Janny et al., 2023)—require reasoning about complex dynamical systems. These systems are naturally modeled as latent functions over continuous domains, yet are observed only through finitely many noisy and irregular measurements. From a probabilistic perspective, such data are generated by an underlying stochastic process. Gaussian processes (GPs; Rasmussen et al. 2006) provide a principled Bayesian framework with closed-form inference and uncertainty estimates, but their cubic computational cost and reliance on carefully designed kernels severely limitscalability, particularly in high-dimensional settings.  \nThe scalability and expressive power of deep neural networks (LeCun et al., 2015) have motivated hybrid approaches that combine deep learning with GP-like probabilistic structure (MacKay, 1995 ; Korshunova et al., 2018 ; Sun et al., 2019 ; Dupont et al., 2021 ; Phillips et al., 2022) . Among these, neural processes (NPs; Garnelo et al. 2018a ;b)—a family of models that learn distributions over functions—have emerged as a particularly influential framework and have since been extended in numerous directions (Jha et al., 2022) . In this work, we focus on conditional NPs (CNPs; Garnelo et al. 2018a), with particular emphasis on translation-equivariant designs (Gordon et al., 2019 ; Huang et al., 2023 ; Ashman et al., 2024a) .  \nTranslation-equivariant NPs build on the convolutional deep set framework of Gordon et al. (2019), which represents equivariant set functions as a functional embedding followed by a continuous translation-equivariant operator. In practice, this opera","cbCaihwAJWlmXOKM","https://ap.wps.com/l/cbCaihwAJWlmXOKM","pdf",2992459,1,46,"English","en",105,"# Abstract\n## Introduction\n## Background and Motivation\n## Problem with Existing Translation-Equivariant NPs\n## Proposed Contributions","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses how to model unknown latent functions using finite, irregularly sampled noisy measurements, aiming for both efficiency and generalization under translation equivariance.\"},{\"question\":\"What are the paper’s two main technical contributions?\",\"answer\":\"It uses a Volterra expansion to approximate continuous translation-equivariant operators with higher-order convolution sums, and it introduces set Fourier convolutions (SFConvs) to parameterize frequency-domain operators directly on irregular points.\"},{\"question\":\"How do the proposed models improve scalability compared with prior approaches?\",\"answer\":\"Convolutional translation-equivariant models require embedding onto dense uniform grids and suffer from grid-resolution costs, while attention-based models scale quadratically; SFConvs instead operate on irregular samples and scale approximately linearly in the number of observations, using approximately global receptive fields.\"}]",1784196149,116,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"revisiting-neural-processes-via-fourier-transform-and-volterra-series","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/revisiting-neural-processes-via-fourier-transform-and-volterra-series/84503/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper address?","Question",{"text":74,"@type":75},"It addresses how to model unknown latent functions using finite, irregularly sampled noisy measurements, aiming for both efficiency and generalization under translation equivariance.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What are the paper’s two main technical contributions?",{"text":79,"@type":75},"It uses a Volterra expansion to approximate continuous translation-equivariant operators with higher-order convolution sums, and it introduces set Fourier convolutions (SFConvs) to parameterize frequency-domain operators directly on irregular points.",{"name":81,"@type":72,"acceptedAnswer":82},"How do the proposed models improve scalability compared with prior approaches?",{"text":83,"@type":75},"Convolutional translation-equivariant models require embedding onto dense uniform grids and suffer from grid-resolution costs, while attention-based models scale quadratically; SFConvs instead operate on irregular samples and scale approximately linearly in the number of observations, using approximately global receptive fields.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]