[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83100-en":3,"doc-seo-83100-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83100,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","On Low-Rank Tensor Train Approximability for Linear Nearest Neighbor Systems","Low-rank tensor methods offer efficient numerical treatment for equations posed on high-dimensional state spaces, including nearest neighbor interaction systems such as the Ising model, Markov jump processes, and 1D finite-state quantum systems. Although tensor train (matrix product state) models enable highly efficient simulations, rigorous justification of required ranks remains challenging. This work derives size-independent low-rank approximability statements by analyzing rank increase in Krylov subspace methods and applying techniques from area-law studies to nearby structured polynomial and dissipative linear ODEs. Numerical experiments support the theory.","arXiv :2607 .06453v1 [math .NA] 7 Jul 2026  \nOn low-rank tensor train approximability for linear  \nnearest neighbor systems  \nPatrick Gelß∗ Sebastian Matera† Reinhold Schneider‡ André Uschmajew§  \nAbstract  \nLow-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model or more general Markov jump processes, as well as 1D finite-state quantum systems are examples of such problems. While low-rank tensor train/matrix product state models have been shown to be highly efficient for the simulation of such systems, providing theoretical justification for this remains a challenging task. One approach for obtaining estimates on required ranks for certain accuracies is to investigate the rank increase in Krylov subspace methods for solving the problem at hand. In the context of area laws for ground states of 1D spin systems, nontrivial results on rank-increasing properties of nearest neighbor operator polynomials have been obtained in work of Arad et al. [arXiv:1301.1162] by studying the partial commutativity of local operators. In the present work, this technique is applied to polynomial methods for definite linear equations and dissipative linear ODEs with nearest neighbor structure. This allows to derive corresponding low-rank approximability statements for solutions of such problems which are independent of the system size. Numerical simulations of high-dimensional nearest neighbor systems illustrate the theoretical findings.  \n1 Introduction  \nWe consider low-rank approximation to solutions of high-dimensional linear equations  \nAu = b (1.1)  \nor linear ordinary differential equations  \n d  \ndt u = Au, u (0) = u0 , (1.2)  \nthat are posed on a d-fold tensor product space,  \nd  \nV := OVµ , (1.3)  \nµ=1  \nwhere Vµ are finite-dimensional K-vector spaces (K = R or K = C) of dimension nµ ≥ 2. Here A is a linear operator on V. Such equations in tensor product spaces arise in several situations, one being the discretization of linear partial differential or integral equations in spaces of multivariate functions. This work, however, is mainly motivated by applications in inverse problems [41, 28 , 51], Markov jump processes [46, 20, 27, 26], and finite-state quantum systems [60, 29, 68] .  \nBy identifying the tensor product space V with the space Kn1 × · ·· ×nd via a fixed tensor product basis, a tensor u ∈ V is identified with a d-dimensional (in case of (1.2) time-dependent) array  \nu  [u (i1 , . . . , id )]  \n∗ AI in Society, Science, and Technology, Zuse Institute Berlin, 14195 Berlin, Germany †Theory Department, Fritz Haber Institute of the Max Planck Society, 14195 Berlin, Germany ‡Institute of Mathematics, Technical University of Berlin, 10623 Berlin, Germany  \n§ Institute of Mathematics & Centre for Advanced Analytics and Predictive Sciences, University of Augsburg, 86159 Augsburg, Germany  \nof coefficients in K indexed via d discrete indices iµ ∈ {1, . . . , nµ } . Since the dimension n 1 · · · nd of the space V grows exponentially with d, the practical representation of its elements, and even more so, the numerical solution of equations like (1.1) or (1.2) defined on that space poses great challenges for large d. This is often referred to as the curse of dimensionality. It is therefore of interest to identify problems which nevertheless allow for efficient numerical treatment because their solution can be approximated using low-parametric (data-sparse) representations.  \nIn the last decades, low-rank tensor techniques have been developed as a powerful tool to deal with high-dimensional problems under suitable structural assumptions on operators and data. The idea of these methods as outlined in seminal works such as [12] is to apply low-parametric representations of higher-order tensors based on suitable low-rank tensor formats. Foundational mathematical aspects are presented in the monographs [36, 50], while compr","cbCaihcAdsuaaPnL","https://ap.wps.com/l/cbCaihcAdsuaaPnL","pdf",1057374,2,1,32,"English","en",105,"# Introduction\n## Low-rank tensor approximation and the curse of dimensionality\n## Tensor train (TT) format and matrix product states\n## Problem setting: linear equations and dissipative linear ODEs","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper studies low-rank approximation of solutions to high-dimensional linear equations and linear ODEs with nearest neighbor structure, focusing on rank requirements in the tensor train (TT) format.\"},{\"question\":\"Why is the tensor train format important in this context?\",\"answer\":\"TT (matrix product state) representations can store tensors with much fewer parameters than full high-dimensional arrays, potentially avoiding the curse of dimensionality when TT ranks remain sufficiently small.\"},{\"question\":\"How are low-rank approximability results obtained?\",\"answer\":\"The approach estimates required ranks by investigating rank growth in Krylov subspace methods and by extending techniques used for rank-increasing properties of nearest neighbor operator polynomials to polynomial methods for structured linear equations and dissipative linear ODEs.\"}]",1784185242,81,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"on-low-rank-tensor-train-approximability-for-linear-nearest-neighbor-systems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-low-rank-tensor-train-approximability-for-linear-nearest-neighbor-systems/83100/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address?","Question",{"text":75,"@type":76},"The paper studies low-rank approximation of solutions to high-dimensional linear equations and linear ODEs with nearest neighbor structure, focusing on rank requirements in the tensor train (TT) format.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the tensor train format important in this context?",{"text":80,"@type":76},"TT (matrix product state) representations can store tensors with much fewer parameters than full high-dimensional arrays, potentially avoiding the curse of dimensionality when TT ranks remain sufficiently small.",{"name":82,"@type":73,"acceptedAnswer":83},"How are low-rank approximability results obtained?",{"text":84,"@type":76},"The approach estimates required ranks by investigating rank growth in Krylov subspace methods and by extending techniques used for rank-increasing properties of nearest neighbor operator polynomials to polynomial methods for structured linear equations and 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