A Geometric Analysis of Quantum-Inspired LocalTensor Regression: Formal Proofs of Convergence and Tractability

A Geometric Analysis of Quantum-Inspired LocalTensor Regression: Formal Proofs of Convergence and Tractability | 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 A Geometric Analysis of Quantum-Inspired LocalTensor Regression: Formal Proofs of Convergence and Tractability Nihal Shetty, Ashil Shetty, B Karthik Udupa This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7917214/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract We present a rigorous mathematical analysis of the Quantum-Inspired Local Tensor Regression (QILTR) framework for modeling non-stationary, multi-linear systems. QILTR's central hypothesis is that weighting local regression problems by the Bures distance—a metric derived from the Quantum Fisher Information (QFI)—improves optimization properties. We formalize this by recasting QILTR as a Riemannian optimization problem on the manifold of quantum states, establishing that the QILTR update approximates a natural gradient descent step measured by the QFI metric. Under local geodesic convexity, we derive a formal convergence rate and demonstrate that QFI-based weighting produces a better-conditioned local Hessian matrix compared to standard Euclidean kernels. We analyze low-rank tensor estimation via weighted Alternating Least Squares (ALS), establishing the regularizing properties of quantum-geometric weights in this non-convex setting, and prove computational tractability with polynomial-time complexity per centroid under dual constraints of low-dimensional quantum encoding and low-rank coefficient tensors. Comprehensive empirical validation on synthetic datasets (150 trials, five scenarios) and real-world quantum chemistry data (QM9, ten trials) confirms that the bounded hyperbolic tangent scaling function prevents outlier amplification, yielding a 4.9× improvement. We transparently report that theoretical convergence rate advantages are not empirically observable, as both QILTR and Euclidean-LTR converge rapidly. Practical benefits manifest as improved solution quality (+1.22% on QM9). By grounding QILTR in information geometry with empirical rigor, we position it as a geometrically optimized and computationally viable scientific machine learning tool with validated design principles and realistic expectations. Quantum-inspired machine learning Information geometry Riemannian optimization Tensor regression Local regression Quantum Fisher Information Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 08 Jan, 2026 Reviewers invited by journal 07 Nov, 2025 Editor assigned by journal 07 Nov, 2025 Submission checks completed at journal 27 Oct, 2025 First submitted to journal 21 Oct, 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7917214","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":535280115,"identity":"dfc791cf-b33e-48b7-b144-25088c906c3b","order_by":0,"name":"Nihal 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QILTR's central hypothesis is that weighting local regression problems by the Bures distance\u0026mdash;a metric derived from the Quantum Fisher Information (QFI)\u0026mdash;improves optimization properties. We formalize this by recasting QILTR as a Riemannian optimization problem on the manifold of quantum states, establishing that the QILTR update approximates a natural gradient descent step measured by the QFI metric. Under local geodesic convexity, we derive a formal convergence rate and demonstrate that QFI-based weighting produces a better-conditioned local Hessian matrix compared to standard Euclidean kernels. We analyze low-rank tensor estimation via weighted Alternating Least Squares (ALS), establishing the regularizing properties of quantum-geometric weights in this non-convex setting, and prove computational tractability with polynomial-time complexity per centroid under dual constraints of low-dimensional quantum encoding and low-rank coefficient tensors. 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