[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84899-en":3,"doc-seo-84899-105":29,"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":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},84899,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","A Unique Normal Form for Tensor Trains over Arbitrary Fields","Tensor trains (matrix-product states) are compressed representations widely used in computer science and physics, recently linked to binary decision diagrams over the 2-element Galois field. This work addresses reducibility and the lack of SVD-based uniqueness by introducing a unique normal form for tensor trains over arbitrary fields. It provides a polynomial-time reduction strategy, methods to extract the normal form directly from a full tensor, retrieve its leading index/value, and bound the size after full reduction relative to naive storage. The approach uses a rank-revealing LDPU decomposition to guarantee uniqueness and strengthen tensor trains as a formal tool.","arXiv :2607 .0627 1v 1 [ cs .DS] 7 Jul 2026  \nA Unique Normal Form for Tensor Trains over Arbitrary Fields  \nRenaud Vilmart \\# 􀀚  \nUniversité Paris-Saclay, Inria, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France.  \n~~ Abstract ~~  \nTensor trains (or Matrix-Product States) are a data structure used in many fields of computer science and physics. They were recently shown to generalise binary decision diagrams when used over the 2-element Galois field, prompting the question of their reducibility in such a context, when the standard approach, over real or complex number, is not amenable to finite fields.  \nWe provide here a unique normal form and associated polynomial-time reduction strategy for tensor trains over arbitrary fields. We also show how to directly extract a normal form out of a full tensor, how to get the leading index and value of a normal form, and an upper bound on the size of a fully-reduced tensor train relative to a naive storage of the full tensor.  \nOn the one hand, this work strengthens the use of tensor trains as a relevant formal tool. On the other hand, from the perspective of tensor networks, it extends the formalism to more general settings than the well-studied real and complex fields, and crucially provides the first tensor train form with the uniqueness property.  \n2012 ACM Subject Classification Theory of computation → Data structures design and analysis; Theory of computation → Equational logic and rewriting  \nKeywords and phrases Tensor trains, Normal form, Rewrite strategy, Uniqueness, Finite fields  \n 1  Introduction  \nTensors are multidimensional extensions of vectors and matrices [12], used pervasively both for modelling physics (continuum mechanics, electromagnetism, general relativity, quantum mechanics, ...), and as a tool for computer science (computer vision, linear system solving, machine learning, quantum computing, ...) .  \nA tensor is a rather big object, the amount of data required to store it depending exponentially in its order (i.e. the number of dimensions; matrices and vectors being respectively order-2 and order-1 tensors) . To deal with this issue, schemes have been devised to compress the data, by decomposing the tensor into a tensor network: a collection of smaller-ordered tensors composed together following a given topology [24] . It is then possible to exploit the Singular Value Decomposition (SVD) to decrease the dimensions between the tensors, either exactly (by exploiting the rank-revealing property of the SVD), or approximately, by chopping off singular values below a given threshold, and hence performing low-rank approximations [25] .  \nA first caveat of the SVD is that it is in general not unique, which means that SVD-based reduction schemes never enjoy a full uniqueness property [12] . The second is that it is only applicable to matrices over R or C (hence using floats in practice) . For most considered applications, like the aforementioned, this is fine. However, it recently was shown that tensor trains (tensor networks consisting of order-3 tensors connected in a line) over F2 could be used to generalise Binary Decision Diagrams (BDDs) [23, 29] . In particular, it has been shown that there exist families of tensor trains over F2 of polynomial size, that would require BDDs of exponential size to represent them [23] .  \nBinary decision diagrams [18] are a data structure used to represent boolean functions, and that make use of the functions’ structure to reduce their size. These diagrams and their  \n2 A Unique Normal Form for Tensor Trains over Arbitrary Fields  \nnumerous variants (Zero-suppressed DDs [22], BDDs with complemented edges [6], ...) can be and have been used for a plethora of applications, such as model checking [11], model counting [9], circuit synthesis [28], regular expression matching [33], multiobjective discrete optimization [4], ..., or simply for the storing of boolean polynomials [7] .  \nThe fact that BDDs a","cbCaigkHcIyRUObs","https://ap.wps.com/l/cbCaigkHcIyRUObs","pdf",625231,1,28,"English","en",105,"# Introduction\n## Motivation from tensor train compression and decision diagrams\n## Background on SVD limitations and the need for uniqueness\n# Core contributions\n## Unique normal form over arbitrary fields\n## Polynomial-time reduction and direct extraction from full tensors\n## LDPU decomposition and rank-revealing reduction mechanism","[{\"question\":\"What problem does the paper address regarding tensor train reducibility?\",\"answer\":\"It targets reducibility and uniqueness when tensor trains are considered over finite fields, where the standard SVD-based reduction approach (over real/complex numbers) does not apply directly.\"},{\"question\":\"What new concept does the paper introduce for tensor trains?\",\"answer\":\"It introduces a unique normal form for tensor trains over arbitrary fields, designed to minimize size and be reachable efficiently.\"},{\"question\":\"How is uniqueness achieved in the proposed reduction strategy?\",\"answer\":\"The method relies on a rank-revealing LDPU decomposition, a variant of LU decomposition, and uses it to define subsequent tensor train normal form and reduction steps with a uniqueness 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problem does the paper address regarding tensor train reducibility?","Question",{"text":75,"@type":76},"It targets reducibility and uniqueness when tensor trains are considered over finite fields, where the standard SVD-based reduction approach (over real/complex numbers) does not apply directly.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What new concept does the paper introduce for tensor trains?",{"text":80,"@type":76},"It introduces a unique normal form for tensor trains over arbitrary fields, designed to minimize size and be reachable efficiently.",{"name":82,"@type":73,"acceptedAnswer":83},"How is uniqueness achieved in the proposed reduction strategy?",{"text":84,"@type":76},"The method relies on a rank-revealing LDPU decomposition, a variant of LU decomposition, and uses it to define subsequent tensor train normal form and reduction steps with a uniqueness 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