[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82420-en":3,"doc-seo-82420-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},82420,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Characterization of the Basin of Convexity for Multi-snapshot Spike Deconvolution via Variable Projection","Multi-snapshot spike deconvolution aims to recover the locations of sparse impulses from noisy convolutions with a known point spread function (PSF) across multiple snapshots. The work employs a variable-projection formulation that eliminates amplitudes in closed form, yielding a nonconvex least-squares problem over spike locations. An explicit characterization of the basin of convexity is given using PSF power spectral density and smoothness, showing how bandwidth and separation affect local geometry. Within this basin, the estimator is consistent with increasing snapshots and enjoys sharper adversarial-noise error bounds via local Lipschitz properties, alongside gradient-descent local convergence. Numerical experiments confirm performance using modified ESPRIT initialization and refinement.","arXiv :2607 .09593v1 [ stat .ML] 10 Jul 2026  \nCharacterization of the basin of convexity for multi-snapshot spike deconvolution via variable  \nprojection  \nMeghna Kalra 1*, Maxime Ferreira Da Costa2 and Kiryung Lee 1  \n1 Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave., Columbus, 43210, OH, United States.  \n2 Laboratory of Signals and Systems, CentraleSup´elec, Universit´e Paris–Saclay, 3 rue Joliot Curie, Gif-sur-Yvette, 91190, France.  \n*Corresponding author(s). E-mail(s): [kalra.42@osu.edu](kalra.42@osu.edu) ; Contributing [authors: maxime.ferreira@centralesupelec.fr](authors: maxime.ferreira@centralesupelec.fr);  \n[kiryung@ece.osu.edu](kiryung@ece.osu.edu) ;  \nAbstract  \nWe study the problem of multi-snapshot spike deconvolution, where the goal is to recover the locations of sparse impulses from their noisy convolution with a known point spread function (PSF) across multiple snapshots. We adopt a variable-projection formulation that eliminates the amplitudes in closed form, reducing the task to a nonconvex least-squares problem over the spike locations alone, which we refer to as the variable-projection formulation of spike deconvolution (VarProSD) . We provide an explicit characterization of the basin of convexity of the VarProSD objective in terms of key PSF properties, including its power spectral density and smoothness, revealing how sampling bandwidth and spike separation influence the local geometry. Within this basin, we establish that the estimator is consistent in the number of snapshots under stochastic noise, and provide a complementary, sharper error bound under adversarial noise via the local Lipschitz property of the inverse map. We further show local convergence guarantees for gradient descent when initialized within the basin. A central ingredient throughout is the use of Beurling–Selberg extremal approximations, which enable sharp, PSF-agnostic bounds on the conditioning of the structured matrices arising in the optimization landscape. Numerical experiments validate our theoretical findings and demonstrate the effectiveness of modified ESPRIT initialization followed by gradient-based refinement.  \n1  \nKeywords: Spike Deconvolution, Variable Projection, Beurling–Selberg Approximation, Local convergence.  \n2  \nContents  \n1 Introduction 4  \n1.1 Problem Formulation ............................ 4  \n1.2 Related Work ................................ 5  \n1.3 Contributions ................................ 6  \n1.4 Mathematical Notation .......................... 10  \n1.5 Organization of the paper ......................... 11  \n2 Main Results 11  \n2.1 Local Neighborhood and Geometry .................... 13  \n2.2 Adversarial Noise Analysis ......................... 15  \n2.2.1 Alternative Stability Bound under Adversarial Noise ...... 16  \n2.2.2 Construction of Worst-Case Adversarial Noise .......... 16  \n2.2.3 Comparison of Error Bounds (Theorem 2 vs. Theorem 3) ... 18  \n2.3 Local Convergence Analysis of Gradient Descent ............ 19  \n3 Numerical Results 20  \n3.1 Initialization methods and sensitivity to bandwidth ........... 21  \n3.2 Optimal bandwidth selection via PSF characteristics .......... 22  \n3.3 Error scaling with key parameters .................... 26  \n3.4 Performance under random and adversarial noise ............ 27  \n4 Key Technical Lemmas about Conditioning of Structured Matrices 28  \n5 Proof of Main Results 35  \n5.1 Local Geometry of the VarProSD Objective within the Basin ..... 35  \n5.2 Proof of Theorem 2 and Theorem 6 .................... 36  \n5.3 Proof of Theorem 3 ............................. 39  \n6 Discussion 41  \nA Prior Art on the Conditioning of Structured Matrices 43  \nB Properties of the Khatri-Rao and Hadamard Products 44  \nC Gradient, Jacobian and Hessian Computations for the VarProSD Objective 45  \nC.1 Proof of Lemma 5 ............................. 46  \nC.2 Jacobian Gramian of the residual ..................... 46  \nC.3 Hessian expression ","cbCairhb0fLj8aet","https://ap.wps.com/l/cbCairhb0fLj8aet","pdf",2124959,1,64,"English","en",105,"# Introduction\n## Problem Formulation\n## Related Work\n## Contributions\n# Main Results\n## Local Neighborhood and Geometry\n## Adversarial Noise Analysis\n## Local Convergence Analysis of Gradient Descent\n# Numerical Results\n## Initialization methods and sensitivity to bandwidth\n## Optimal bandwidth selection via PSF characteristics\n## Error scaling with key parameters\n## Performance under random and adversarial noise\n# Key Technical Lemmas about Conditioning of Structured Matrices\n# Proof of Main Results\n## Local Geometry of the VarProSD Objective within the Basin\n## Proofs of Main Theorems\n# Discussion","[{\"question\":\"What problem does the paper address in multi-snapshot spike deconvolution?\",\"answer\":\"It studies how to recover the common spike locations from multiple noisy observations, where each snapshot is the convolution of sparse impulses with a known PSF.\"},{\"question\":\"How does variable projection simplify the optimization?\",\"answer\":\"It removes the amplitudes analytically, reducing the task to a nonconvex least-squares problem that depends only on the spike locations.\"},{\"question\":\"What determines the basin of convexity in the proposed formulation?\",\"answer\":\"The basin is characterized explicitly using PSF properties, especially the PSF power spectral density and smoothness, and it depends on sampling bandwidth and spike separation.\"}]",1784180268,161,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"characterization-of-the-basin-of-convexity-for-multi-snapshot-spike-deconvolution-via-variable-projection","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/characterization-of-the-basin-of-convexity-for-multi-snapshot-spike-deconvolution-via-variable-projection/82420/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper address in multi-snapshot spike deconvolution?","Question",{"text":74,"@type":75},"It studies how to recover the common spike locations from multiple noisy observations, where each snapshot is the convolution of sparse impulses with a known PSF.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does variable projection simplify the optimization?",{"text":79,"@type":75},"It removes the amplitudes analytically, reducing the task to a nonconvex least-squares problem that depends only on the spike locations.",{"name":81,"@type":72,"acceptedAnswer":82},"What determines the basin of convexity in the proposed formulation?",{"text":83,"@type":75},"The basin is characterized explicitly using PSF properties, especially the PSF power spectral density and smoothness, and it depends on sampling bandwidth and spike 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