Reducing swamp behavior for the canonical polyadic decomposition problem by rank-1 freezing

Reducing swamp behavior for the canonical polyadic decomposition problem by rank-1 freezing | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Reducing swamp behavior for the canonical polyadic decomposition problem by rank-1 freezing Charlotte Vermeylen, Nico Vervliet, Lieven De Lathauwer This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6234994/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 05 Jun, 2025 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract A novel optimization framework is proposed for solving the low-rank tensor approximation problem using the canonical polyadic decomposition (CPD). This can be a difficult optimization problem for certain tensors, e.g., due to degeneracy, i.e., a tensor that can be approximated arbitrarily closely by an ill-conditioned tensor of lower rank. This is one of the phenomena that are encountered in regions of slow convergence, informally known as swamps. Numerical experiments with state-of-the-art optimization algorithms indicate that in a swamp often only a few rank-1 terms are modified while others stagnate. Often, the non-stagnant terms are problematic and form an ill-conditioned decomposition. To address this, we propose to temporarily freeze the stagnant terms. The lower number of terms has several benefits: it simplifies the problem by reducing the number of variables and reduces the cost per iteration significantly. Furthermore, in many cases the residual tensor can be compressed, and an algebraic (re)initialization can be carried out, even if this was not possible for the original tensor. A refinement step can further improve the accuracy if desired. We provide theoretical insights of why terms can stagnate. More specifically, we prove that terms that are close to the solution are not modified anymore in further optimization steps under certain assumptions. Extensive numerical experiments show that our framework greatly facilitates escaping from swamps. The resulting algorithm outperforms current state-of-the-art approaches on difficult-to-decompose tensors, both in accuracy and computation time, and has similar performance on easier problems. tensors canonical polyadic decomposition numerical optimization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 05 Jun, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 08 May, 2025 Reviews received at journal 29 Apr, 2025 Reviews received at journal 07 Apr, 2025 Reviewers agreed at journal 25 Mar, 2025 Reviewers agreed at journal 25 Mar, 2025 Reviewers invited by journal 25 Mar, 2025 Editor assigned by journal 17 Mar, 2025 Submission checks completed at journal 17 Mar, 2025 First submitted to journal 15 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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This can be a difficult optimization problem for certain tensors, e.g., due to degeneracy, i.e., a tensor that can be approximated arbitrarily closely by an ill-conditioned tensor of lower rank. This is one of the phenomena that are encountered in regions of slow convergence, informally known as swamps. Numerical experiments with state-of-the-art optimization algorithms indicate that in a swamp often only a few rank-1 terms are modified while others stagnate. Often, the non-stagnant terms are problematic and form an ill-conditioned decomposition. To address this, we propose to temporarily freeze the stagnant terms. The lower number of terms has several benefits: it simplifies the problem by reducing the number of variables and reduces the cost per iteration significantly. Furthermore, in many cases the residual tensor can be compressed, and an algebraic (re)initialization can be carried out, even if this was not possible for the original tensor. 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