[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85476-en":3,"doc-seo-85476-105":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},85476,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Stochastic Zeroth-Order Method for Computing Generalized Rayleigh Quotients","Maximizing the (generalized) Rayleigh quotient is a core task in numerical linear algebra. Conventional methods often depend on matrix–adjoint products and can be highly sensitive to errors caused by adjoint mismatches, while also requiring inverse-related operations for generalized settings. The paper presents a stochastic zeroth-order Riemannian algorithm that avoids both adjoint and matrix-inverse computations, uses only products with A and B, and comes with convergence guarantees.","Stochastic Zeroth-Order Method for Computing Generalized Rayleigh Quotients  \narXiv :2512 .05520v2 [math .OC] 11 Jul 2026  \nJonas Bresch∗  \n[bresch@math.tu-berlin.de](bresch@math.tu-berlin.de)  \nOleh Melnyk∗ [melnyk@math.tu-berlin.de](melnyk@math.tu-berlin.de)[ ](melnyk@math.tu-berlin.de)Gabriele Steidl∗ [steidl@math.tu-berlin.de](steidl@math.tu-berlin.de)  \nMartin Schoen∗  \nJuly 14, 2026  \nAbstract  \nThe maximization of the (generalized) Rayleigh quotient is a central problem in numerical linear algebra. Conventional algorithms for its computation typically rely on matrix–adjoint products, making them sensitive to errors arising from adjoint mismatches. To address this issue, we introduce a stochastic zeroth-order Riemannian algorithm that maximizes the generalized Rayleigh quotient without requiring adjoint or matrix inverse computations. We provide theoretical convergence guarantees showing that the iterates converge to the set of global maximizers of the (generalized) Rayleigh quotient and the norm of the Riemannian gradient vanishes at a sublinear rate with probability one. Our theoretical results are supported by numerical experiments, which demonstrate the excellent performance of the proposed method compared to state-of-the-art algorithms.  \nKeywords. generalized Rayleigh quotient · spectral norm · stochastic optimization · zeroth-order optimization · Riemannian optimization  \nMSC. 58C40 · 65F15 · 65F35 · 15A60 · 68W20  \n1. Introduction  \nIn this paper, we are interested in the maximization of the generalized Rayleigh quotient  \n⟨v, Av⟩  \nR (A, B) = max (1)  \nv∈Rd \\{0} ⟨v, Bv⟩  \nwithout explicitly using the inverse of the positive definite B ∈ Rd×d or the transpose of A ∈ Rd×d. The maximization of (1) is a fundamental problem in various applications  \n∗ Technische Universit¨at Berlin, Straße des 17 . Juni 136, Berlin, 10587, Germany  \nsuch as spectral equivalence of the matrices A and B [1], generalized singular value [2], and tensor [3] decompositions. For the identity matrix B, the Rayleigh quotient is also known as the numerical abscissa and is used for the stability analysis of nonsymmetric matrices in partial differential equations [4, 5, 6] .  \nThe maximization of (1) can be performed by a number of methods. Classical iterative schemes for real eigenvalue problems use Rayleigh quotient iterations [7, 8, 9, 10] and their block [11], Krylov-style [12], or matrix-free [13] variants. A more robust class of approaches is based on the min–max characterizations for generalized eigenvalue problems [14, 15, 16] . Alternatively, maximization of (1) can be performed using Riemannian optimization techniques [17, 18, 19] . Another class of algorithms relies on constructing rank-one perturbations of A leading to a stable approximation of the largest eigenvalue [20] . These can be prohibitively expensive for large-scale matrices, and a more scalable algorithmic approach based on subspace methods was proposed in [21, 22] . Sketching methods [23, 24, 25] solve the problem using random dimensional reduction techniques. All the above methods rely on matrix–adjoint product and/or require access to the inverse of B.  \nThe motivation for developing inverse-and adjoint-free methods stems from two main considerations. First, computing B −1 or performing matrix–vector products involving it, is computationally expensive and susceptible to numerical inaccuracies. Second, in imaging applications such as computed tomography [26, 27, 28, 29], the transpose AT is often replaced by an approximate, but computationally tractable operator. This substitution introduces what is known as adjoint mismatch, which can lead to significant reconstruction errors.  \nWhen AT and B −1 are unavailable, zeroth-order optimization methods can be employed, as they rely solely on evaluation of products with A and B. These methods approximate the gradient using finite-difference schemes [30, 31, 32] . In the context of Rayleigh quotient optimization, the bundled gra","cbCaif5T8xxJ0nPq","https://ap.wps.com/l/cbCaif5T8xxJ0nPq","pdf",1770095,1,38,"English","en",105,"# Introduction\n## Motivation and problem setting\n## Related work and limitations\n## Proposed approach and contributions\n# Outline of the Paper\n## Preliminaries\n## Algorithm presentation\n## Termination and convergence analysis\n## Extensions and experiments\n# Proofs and technical lemmas","[{\"question\":\"What optimization problem does the method address?\",\"answer\":\"It targets the maximization of the generalized Rayleigh quotient R(A,B)=⟨v,Av⟩/⟨v,Bv⟩ over nonzero vectors v in Rd.\"},{\"question\":\"Why does the paper avoid computing B^{-1} and matrix transposes?\",\"answer\":\"Computing B^{-1} is expensive and error-prone, and in imaging contexts the transpose A^T is often replaced by an approximate operator, which creates adjoint mismatch and can cause reconstruction errors.\"},{\"question\":\"What guarantees are provided for the proposed stochastic zeroth-order Riemannian algorithm?\",\"answer\":\"The paper proves that the iterates converge to the set of global maximizers of the generalized Rayleigh quotient and that the Riemannian gradient norm vanishes at a sublinear rate with probability 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optimization problem does the method address?","Question",{"text":75,"@type":76},"It targets the maximization of the generalized Rayleigh quotient R(A,B)=⟨v,Av⟩/⟨v,Bv⟩ over nonzero vectors v in Rd.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does the paper avoid computing B^{-1} and matrix transposes?",{"text":80,"@type":76},"Computing B^{-1} is expensive and error-prone, and in imaging contexts the transpose A^T is often replaced by an approximate operator, which creates adjoint mismatch and can cause reconstruction errors.",{"name":82,"@type":73,"acceptedAnswer":83},"What guarantees are provided for the proposed stochastic zeroth-order Riemannian algorithm?",{"text":84,"@type":76},"The paper proves that the iterates converge to the set of global maximizers of the generalized Rayleigh quotient and that the Riemannian gradient norm vanishes at a sublinear rate with probability 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