[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81512-en":3,"doc-seo-81512-105":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},81512,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","On Trimming Tensor-structured Measurements and Efficient Low-rank Tensor Recovery","Efficient low-rank tensor recovery is studied under linear measurements with tensorial (memory-efficient) structure. Standard iterative recovery guarantees rely on norm-preserving operator assumptions such as the restricted isometry property (RIP), yet tensor-structured random maps fail to preserve point-wise geometry well enough. To remedy this, local trimming techniques are proposed to restore geometry-preservation properties for tensor-structured measurements. Two new tensor IHT variants are introduced—adaptive gradient trimming and a randomized Kaczmarz-based method—supported by initial theoretical guarantees and experiments on real and synthetic data, demonstrating improved efficiency versus TIHT for low HOSVD- and CP-rank tensors.","arXiv :2502 .02843v2 [math .NA] 9 Jul 2026  \nON TRIMMING TENSOR-STRUCTURED MEASUREMENTS AND EFFICIENT LOW-RANK TENSOR RECOVERY  \nSHAMBHAVI SURYANARAYANAN, ELIZAVETA REBROVA  \nAbstract. In this paper, we take a step towards developing efficient hard thresholding methods for low-rank tensor recovery from linear measurements with tensorial structure. Theoretical guarantees for many standard iterative low-rank recovery methods, such as iterative hard thresholding (IHT), are based on model assumptions on the measurement operator, like the restricted isometry property (RIP) . However, tensor-structured random linear maps – while memory-efficient and convenient to apply – lack good restricted isometry properties; that is, they do not preserve the norms of low-rank tensors sufficiently well. To address this, we propose local trimming techniques that provably restore point-wise geometry-preservation properties of tensor-structured maps, making them comparable to those of unstructured linear measurements.  \nThen, we propose two novel versions of tensor IHT algorithms: an adaptive gradient trimming algorithm and a randomized Kaczmarz-based IHT algorithm that efficiently recover low-rank tensors from linear measurements. We provide initial theoretical guarantees for the proposed methods and present numerical experiments on real and synthetic data, highlighting their efficiency over the original TIHT for low HOSVD-and CP-rank tensors.  \nKeywords. Low-rank recovery, tensor-structured data, memory-efficient linear measurements  \nMSC codes. 97N40, 15A69, 15A83, 15B52  \n1. Introduction  \nTensors, as multi-modal arrays, are natural choices for analyzing realistic, high-dimensional structures in various application areas. They have been widely used in recent years for modeling objects in signal processing, medical imaging, machine learning, and other domains [39, 23, 36, 37, 32] . The ubiquity of their applications has motivated the development of specialized techniques to efficiently process large-scale multi-modal data [17, 35, 33] . Due to the large scale of many applications, it is not surprising that the cornerstone techniques in this suite are related to compression and subsequent recovery of tensorial data [47, 29] .  \nData-oblivious random sketching is a fundamental and powerful linear approach for taming largescale data. For a given large instance of the data X ∈ RN, the goal is to replace X with AX where A ∈ Rm×N is a random matrix with m ≪ N. Unlike data-aware low-parametric fitting (such as low-rank fitting) , which aims to find the best possible approximation for a single large and highdimensional object, selecting a random operator A enables us to obtain uniform guarantees for many such large data instances, all while avoiding costly fitting procedures. For example, when it is crucial to guarantee the validity of a data sketch through an iterative process, one requires uniform guarantees over all possible iterates.  \nA key question in oblivious linear dimensionality reduction is how to design memory-efficient random measurement maps A that can be applied to the data fast and lead to good compression with high probability [20, 14] . Memory efficiency is especially important for tensorial data, since vectorizing a tensor of dimension n in d modes leads to an object in RN with N = nd , and sketching matrices have to be correspondingly very large-scale. Recent work has studied tensor-specific, fast, and memory-efficient sketching operators. Some key examples include Kronecker (or, modewise) measurements, face-splitting measurements (that consist of independent scalar Kronecker-structured measurements), and more sophisticated tensor-structured measurements are usually built upon the former (an incomplete list includes [61, 52, 60]) . In addition to requiring fewer random bits to  \n2 SHAMBHAVI SURYANARAYANAN, ELIZAVETA REBROVA  \nstore structured matrices, these strategies allow fast application to tensor data by mimicking its structure. ","cbCaionrpBPv5m6p","https://ap.wps.com/l/cbCaionrpBPv5m6p","pdf",2322419,4,1,34,"English","en",105,"# Introduction\n## Memory-efficient tensor sketching and dimensionality reduction\n## Geometry preservation, RIP, and iterative recovery\n## Low-rank tensor recovery under tensor-structured measurements","[{\"question\":\"Why do standard iterative low-rank recovery guarantees not directly apply to tensor-structured random measurements?\",\"answer\":\"They typically require restricted isometry–type norm preservation (e.g., RIP), while tensor-structured maps do not preserve norms of low-rank tensors sufficiently well, especially in point-wise geometry.\"},{\"question\":\"What are local trimming techniques used for in this work?\",\"answer\":\"They are introduced to provably restore point-wise geometry-preservation properties of tensor-structured measurement maps, making them comparable to unstructured linear measurements.\"},{\"question\":\"What new algorithms are proposed for tensor IHT and what distinguishes them?\",\"answer\":\"The paper proposes an adaptive gradient trimming algorithm and a randomized Kaczmarz-based tensor IHT algorithm, both aimed at efficient recovery of low-rank tensors from structured linear measurements.\"}]",1784173912,86,{"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-trimming-tensor-structured-measurements-and-efficient-low-rank-tensor-recovery","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/on-trimming-tensor-structured-measurements-and-efficient-low-rank-tensor-recovery/81512/",{"url":52,"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},"Why do standard iterative low-rank recovery guarantees not directly apply to tensor-structured random measurements?","Question",{"text":75,"@type":76},"They typically require restricted isometry–type norm preservation (e.g., RIP), while tensor-structured maps do not preserve norms of low-rank tensors sufficiently well, especially in point-wise geometry.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are local trimming techniques used for in this work?",{"text":80,"@type":76},"They are introduced to provably restore point-wise geometry-preservation properties of tensor-structured measurement maps, making them comparable to unstructured linear measurements.",{"name":82,"@type":73,"acceptedAnswer":83},"What new algorithms are proposed for tensor IHT and what distinguishes them?",{"text":84,"@type":76},"The paper proposes an adaptive gradient trimming algorithm and a randomized Kaczmarz-based tensor IHT algorithm, both aimed at efficient recovery of low-rank tensors from structured linear measurements.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]