[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84385-en":3,"doc-seo-84385-105":28,"detail-sidebar-cat-0-en-105":89},{"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":20,"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":11},84385,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Spectral Stability of Pseudoinverse-Based Extreme Learning Machine","Extreme Learning Machine (ELM) obtains output weights analytically via the Moore–Penrose pseudoinverse, yet training speed does not guarantee numerical stability. This work analyzes pseudoinverse-based ELM through its spectral properties by linking the hidden-layer matrix’s smallest singular value to perturbation amplification in the output weights and using the condition number as a quantitative indicator of hidden-layer instability. SVD-based pseudoinverse computation is compared with iterative hyperpower methods, and experiments on synthetic matrices and ELM benchmarks validate the singular-value-driven stability mechanism.","Spectral Stability of Pseudoinverse-Based Extreme Learning Machine  \n2026  \nNguyen Bich Van 1 ,∗ , Khong Ngoc Anh2  \n1 ,2 : Institute for Artificial Intelligence, VNU University of Engineering and Technology, Vietnam  \n1 : [nbvan@vnu.edu.vn](nbvan@vnu.edu.vn) ,2 : [22022549@vnu.edu.vn](22022549@vnu.edu.vn)  \nAbstract  \nExtreme Learning Machine (ELM) computes output weights analytically using the Moore–Penrose pseudoinverse. Although this leads to fast training, its numerical stability depends strongly on the conditioning of the hidden-layer matrix. This paper studies pseudoinverse-based ELM from a spectral perspective. We show that the smallest singular value governs perturbation amplification in the output weights, while the condition number provides a quantitative measure of hidden-layer instability. We compare SVD-based pseudoinverse computation with iterative hyperpower methods and discuss width-dependent conditioning through a random feature interpretation. Experiments on synthetic matrices and ELM benchmarks show that SVD-based methods remain the most reliable under ill-conditioning, while iterative methods are more sensitive to spectral properties. The results suggest that ELM stability is fundamentally governed by the singular-value structure of the hidden-layer matrix.  \n[3], is a single-hidden-layer feedforward which hidden-layer parameters are ran-  \n∗ Corresponding author  \nRelated work on ELM has emphasized fast analytical training and classification performance [3] . The theory of generalized inverses and least-squares solutions is classical [1], while SVD-based conditioning and stability analysis are standard topics in numerical linear algebra [2, 6] . Iterative inverse and pseudoinverse methods, including Newton– Schulz and hyperpower iterations, may be computationally attractive but can be sensitive to initialization and illconditioning. In parallel, random matrix theory provides useful estimates for the singular values of random feature matrices [4, 5] .  \nThe contribution of this paper is a compact spectral interpretation of pseudoinverse-based ELM stability. We connect:  \nhidden-layer spectrum → conditioning → pseudoinverse sensitivity → learning stability.  \nWe also summarize experiments showing how singular values, condition numbers, and solver behavior are related in synthetic matrices and ELM benchmarks.  \n2. Spectral Stability Analysis  \nLet H ∈ RN ×L be the hidden-layer matrix of an ELM and suppose its compact singular value decomposition is  \nH = Ur ΣrVrT , (3)  \nwhere r = rank(H), Ur ∈ RN ×r , Vr ∈ RL ×r have orthonormal columns, Σr = diag(σ1 ,...,σr ) and σmax (H) ≥ σ 1 ≥· · · ≥ σr = σmin (H) > 0 are positive singular values of H. The Moore–Penrose pseudoinverse is  \nH + = Vr Σ−r1 UTr . (4)  \nHence,  \n∥H+ ∥2 = σmi1n(H) . (5)  \nEquation (5) shows that the smallest singular value is the key spectral quantity controlling pseudoinverse amplification.  \nThe condition number of H is  \nσmax (H)  \nκ (H) = (6)  \nσmin (H) .  \nIt measures how unevenly H stretches different directions. If κ (H) is large, at least one direction is nearly collapsed by H , making the least-squares solution sensitive to perturbations.  \nConsider a perturbation T →7 T +∆T. The corresponding perturbation in the ELM output weights is  \n∆β = H + ∆T. (7)  \nTherefore,  \n∥∆β∥2 ≤ ∥H+ ∥2 ∥∆T∥2 =  ∥∆T∥2  (8)  \nσmin (H) .  \nThus, small σmin (H) or large κ (H) implies stronger perturbation amplification. This provides the main theoretical explanation for instability in pseudoinverse-based ELM.  \n3. Solvers and Width-Dependent Conditioning  \nSVD-based pseudoinverse computation directly uses the singular values of H and is typically backward stable because it relies on orthogonal transformations [2, 6] which preserve Euclidean norm. It also allows small singular directions tobe detected and truncated if necessary.  \nIterative methods approximate the pseudoinverse through repeated matrix updates. A representative example is the Newton–Schulz ite","cbCaievVcWnls4dV","https://ap.wps.com/l/cbCaievVcWnls4dV","pdf",220751,3,1,"English","en",105,"# Abstract\n# Spectral Stability Analysis\n# Solvers and Width-Dependent Conditioning\n# Experimental Results","[{\"question\":\"Why does pseudoinverse-based ELM depend on the hidden-layer matrix spectrum?\",\"answer\":\"The analysis shows that perturbation amplification in the learned output weights is governed by the smallest singular value of the hidden-layer matrix, so spectral conditioning directly impacts stability.\"},{\"question\":\"How do SVD-based and iterative pseudoinverse solvers differ in stability?\",\"answer\":\"SVD-based pseudoinverse computation uses singular values and is typically backward stable due to orthogonal transformations, while iterative methods’ convergence can become sensitive to initialization and ill-conditioning.\"},{\"question\":\"What role do random-feature and width effects play in conditioning?\",\"answer\":\"Viewing the hidden-layer matrix as a random feature matrix motivates width-dependent interpretation: larger sample size can improve stability, whereas excessive hidden width may worsen conditioning through its influence on the singular-value structure.\"}]",1784195233,{"code":4,"msg":29,"data":30},"ok",{"site_id":24,"language":23,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":27},"spectral-stability-of-pseudoinverse-based-extreme-learning-machine","",{"@graph":34,"@context":83},[35,51,66],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,48],{"item":39,"name":40,"@type":41,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":20},"https://docshare.wps.com/document/research-report/",{"item":49,"name":13,"@type":41,"position":50},"https://docshare.wps.com/document/spectral-stability-of-pseudoinverse-based-extreme-learning-machine/84385/",4,{"url":49,"name":13,"@type":52,"author":53,"headline":13,"publisher":55,"fileFormat":58,"inLanguage":23,"description":14,"dateModified":59,"datePublished":60,"encodingFormat":58,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":54},"Person",{"url":39,"name":56,"@type":57},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":63,"interactionType":64,"userInteractionCount":20},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"Why does pseudoinverse-based ELM depend on the hidden-layer matrix spectrum?","Question",{"text":73,"@type":74},"The analysis shows that perturbation amplification in the learned output weights is governed by the smallest singular value of the hidden-layer matrix, so spectral conditioning directly impacts stability.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"How do SVD-based and iterative pseudoinverse solvers differ in stability?",{"text":78,"@type":74},"SVD-based pseudoinverse computation uses singular values and is typically backward stable due to orthogonal transformations, while iterative methods’ convergence can become sensitive to initialization and ill-conditioning.",{"name":80,"@type":71,"acceptedAnswer":81},"What role do random-feature and width effects play in conditioning?",{"text":82,"@type":74},"Viewing the hidden-layer matrix as a random feature matrix motivates width-dependent interpretation: larger sample size can improve stability, whereas excessive hidden 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