[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85984-en":3,"doc-seo-85984-105":30,"detail-sidebar-cat-0-en-105":88},{"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":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85984,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization","Recovery of a pair of sparse vectors from few nonlinear measurements of their superposition is studied under incoherent orthonormal bases, an unknown sparse decomposition, and noise that can be heavy-tailed or contaminated. A Huberized data-fidelity with generalized folded-concave penalties (SCAD, MCP) is combined with a two-block proximal alternating algorithm with backtracking (NLD-PALM). The iterates converge to critical points with KL-based guarantees and local linear rates. Statistical results include restricted strong convexity, estimation error bounds, an oracle rate without log factors, and co-equal recovery for unknown monotone links. Experiments show earlier phase transitions than convex demixing.","Demixing Sparse Signals from Nonlinear Observations using Generalized  \nNon-convex Regularization  \nRaziyeh Takbiri∗  \narXiv :2607 . 10618v1 [ stat .ML] 12 Jul 2026  \nAbstract  \nWe consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition: yi = g (⟨ai , Φw ∗ + Ψz ∗ ⟩) + ei , i = 1 , . . . , m, with m ≪ n, incoherent orthonormal bases Φ , Ψ, a scalar link g, and noise ei that may be heavy-tailed or contaminated. We propose a regularization-based framework combining a Huberized data fidelity with generalized folded-concave penalties (SCAD, MCP), and a two-block proximal alternating algorithm with backtracking (NLD-PALM) whose whole iterate sequence provably converges to critical points under the Kurdyka– Lojasiewicz property, with local linear rates. On the statistical side we establish restricted strong convexity of the Huberized nonlinear loss through an exact sign-definite decomposition, and derive estimation error bounds of order σ ps log(n)/m that hold at every localized stationary point, an oracle rate σ ps/m free of log n and shrinkage bias under a beta-min condition, and a co-equal recovery theorem for unknown monotone links via a linear surrogate and a clipped Plan–Vershynin decoupling. The estimator requires no knowledge of the sparsity levels, and its guarantees hold under symmetric noise with only finite variance. Experiments at n = 512 under a frozen datadriven regularization rule show an earlier phase transition than convex ℓ 1 demixing and greedy hard-thresholding baselines, a 35 × accuracy advantage over squared-loss estimation under 5% gross outliers, and successful demixing of spike-plus-background signals observed through a saturating amplifier.  \nKeywords: sparse demixing, nonlinear observations, non-convex regularization, folded-concave penalties, robust recovery, proximal alternating minimization, Kurdyka–Lojasiewicz.  \n1 Introduction  \nSignal demixing — separating a superposition of structured components from few measurements — and nonlinear compressed sensing have each been studied extensively, but their intersection remains sparse. Acquisition  \n∗ R. Takbiri is with the School of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran (e-mail: [raziyeh.takbiri@gmail.com](raziyeh.takbiri@gmail.com)). Code to reproduce all experiments is available with this manuscript.  \nfront ends saturate, clip, compand, and quantize; the recorded data are then nonlinear functions of a mixture. Formally we observe  \nyi = g 􀀀⟨ai , x∗ ⟩ 􀀁 + ei , x∗ = Φw ∗ + Ψz ∗ , (1)  \ni = 1 , . . . , m ≪ n, with ∥w ∗ ∥0 ≤ s 1 , ∥z ∗ ∥0 ≤ s2 , and ask for both components.  \nThe state of the art for (1) is the greedy line of Soltani and Hegde [1, 2]: demixing hard thresholding with sample complexity m = O (slog(n/s)), s = s 1 + s2 , requiring the exact sparsity levels as input and offering no robustness theory. In the linear regime (g = id), non-convex regularization is known to outperform convex ℓ 1 demixing [3, 4], reflecting the general folded-concave phenomenon [5, 6]: penalties that debias large coefficients achieve oracle behavior where the LASSO carries an irreducible λ √ s shrinkage bias [7, 8] . For single-index models without superposition, Plan and Vershynin [9] showed that unknown monotone nonlinearities can be absorbed into an effective scaling plus noise. To the best of our knowledge, no prior work combines generalized non-convex regularization with nonlinear demixing, in either the known-or unknown-link regime.  \n1.1 Contributions  \n1. Estimator (Sec. 2) . A Huberized non-convex program for (1) . Huberization is structural, not cosmetic: it eliminates a curvature–noise compatibility condition that squared-loss analysis provably requires, and extends all guarantees to symmetric noise with only finite variance.  \n2. Algorithm (Sec. 3) . NLD-PALM, a two-block proximal-gradient method with per-block backtracking and an over-relaxation factor η","cbCaiozEdmm2WuLM","https://ap.wps.com/l/cbCaiozEdmm2WuLM","pdf",657153,5,1,7,"English","en",105,"# Introduction\n## Contributions\n# Problem Formulation\n## Model and assumptions","[{\"question\":\"What measurement model and assumptions are used for sparse demixing?\",\"answer\":\"Measurements follow yi = g(⟨ai, Φw* + Ψz*⟩) + ei with m ≪ n, incoherent orthonormal bases Φ and Ψ, and noise ei independent of {ai}. Assumptions include Gaussian sensing, incoherence between bases, and symmetric noise that is either sub-Gaussian or only requires finite variance (allowing heavy tails/contamination).\"},{\"question\":\"What guarantees are provided for the algorithm and the estimation error?\",\"answer\":\"The NLD-PALM algorithm generates a whole-iterate sequence that converges to critical points under the Kurdyka–Łojasiewicz property, with local linear (R-linear) rates. The analysis also establishes restricted strong convexity of the Huberized nonlinear loss and provides estimation error bounds at localized stationary points, including an oracle-rate result under a beta-min condition and a co-equal recovery theorem for unknown monotone links.\"}]",1784207566,18,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":83,"head_meta":85,"extra_data":87,"updated_unix":28},"demixing-sparse-signals-from-nonlinear-observations-using-generalized-non-convex-regularization","",{"@graph":36,"@context":82},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/demixing-sparse-signals-from-nonlinear-observations-using-generalized-non-convex-regularization/85984/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78],{"name":73,"@type":74,"acceptedAnswer":75},"What measurement model and assumptions are used for sparse demixing?","Question",{"text":76,"@type":77},"Measurements follow yi = g(⟨ai, Φw* + Ψz*⟩) + ei with m ≪ n, incoherent orthonormal bases Φ and Ψ, and noise ei independent of {ai}. Assumptions include Gaussian sensing, incoherence between bases, and symmetric noise that is either sub-Gaussian or only requires finite variance (allowing heavy tails/contamination).","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What guarantees are provided for the algorithm and the estimation error?",{"text":81,"@type":77},"The NLD-PALM algorithm generates a whole-iterate sequence that converges to critical points under the Kurdyka–Łojasiewicz property, with local linear (R-linear) rates. The analysis also establishes restricted strong convexity of the Huberized nonlinear loss and provides estimation error bounds at localized stationary points, including an oracle-rate result under a beta-min condition and a co-equal recovery theorem for unknown monotone links.","https://schema.org",{"og:url":52,"og:type":84,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":86,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":89},[90,94,98,102,106,111,115,118,123,126,130],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Comic",60,"comic",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},6,"Technology",50,"technology",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":116,"slug":117},30,"research-report",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":121,"slug":122},9,"Religion & Spirituality",20,"religion-spirituality",{"id":121,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":121,"slug":125},"World Cup","world-cup",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":127,"slug":129},10,"Lifestyle","lifestyle",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":20,"slug":133},19,"General","general"]