[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84943-en":3,"doc-seo-84943-105":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},84943,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Near-Optimal Lower Bounds on One-Bit Compressed Sensing of Approximately Sparse Signals","This paper establishes the first near-optimal lower bounds for one-bit compressed sensing of approximately sparse signals modeled as lying in a scaled ℓ1 ball, a standard relaxation of the exact k-sparse assumption. The work strengthens information-theoretic arguments by constructing pairs of signals inside a small Euclidean ball that remain well separated in ℓ2 distance (up to log factors) while producing indistinguishable binary measurements, extending the results to properly scaled ℓq balls (q in [0,1]).","arXiv :2607 .06750v 1 [ cs .IT] 7 Jul 2026  \nNear-Optimal Lower Bounds on One-Bit Compressed Sensing of  \nApproximately Sparse Signals  \nJunren Chen∗ Arya Mazumdar† Ming Yuan‡  \nJuly 9, 2026  \nAbstract  \nThis paper provides the first near-optimal lower bounds for one-bit compressed sensing of approximately sparse signals lying in a scaled ℓ 1 ball, which is a commonly adopted relaxation of the exactly k-sparse assumption. In prior works, the best known upper bounds on uniform EGcoucaumlispdsreaianessneemdrrasortrenicsaresineg, ofwmoeoreddseetlrabas(h(kth/mneeau)1rln/3yifo)mr,watmlhcyehreindimgthleiors tweedher bmnooduuembndlserfOoorurf mebotargahusutmreheenmtecintans tsoo.nUicfirndalsteoernemsb-ubbeitda small Euclidean ball into the signal set, which is straightforward for the dithered model but relies on a lifting map for the canonical model, and then construct two signals in this small ball that are separated in Euclidean distance by at least (k/m)1/3 (up to logarithmic factor) but are indistinguishable from the binary measurements. Moreover, our argument extends to approximately sparse signals that live in a properly scaled ℓq ball (q ∈ [0 , 1]), yielding a lower bb(oqitd1ip)iFn( g(kin, /mallmya)t,  )drthiscecatuovsesms try,hoeaoenthxtdlyenchbsiarrioadgnscteoersfitozehuerthlcoeawtsererasbnoosifutienoxdnasctttoosstpuhaber-sWnoityeni-bs(qupla0atc)raaniceseds, ℓ 1advsparsersariityal  \n1 Introduction  \nIn one-bit compressed sensing (1bCS), one seeks to recover sparse vectors from the signs of linear measurements [4, 16 , 25 , 26], that is, to recover some x ∈ Σnk ,∗ := Σnk ∩ Sn−1 from  \ny = sign(Ax) (1)  \nand a known measurement matrix A ∈ Rm ×n. Here, Σnk = {u ∈ Rn : ∥u∥0 ≤ k} denotes the cone of k-sparse vectors in Rn , Sn−1 = {u ∈ Rn : ∥u∥2 = 1} denotes the unit Euclidean sphere, and sign(a) := 1 (a ≥ 0)−1(a \u003C 0) is applied element-wise to vectors. Note that it is standard to assume x ∈ Sn−1 in 1bCS since the quantization loses all the information of signal norm. For arbitrary matrix A, any estimator ˆx that is a measurable function of (A, y) admits a uniform recovery ℓ2 error lower  \n∗ Department of Statistics, Columbia University. (Email: [jc6315@columbia.edu](jc6315@columbia.edu))†Halıcıoğlu Data Science Institute, UC San Diego. (Email: [arya@ucsd.edu](arya@ucsd.edu))‡Department of Statistics, Columbia University. (Email: [my2550@columbia.edu](my2550@columbia.edu))  \nbound  \nsup ∥ˆx − x∥2 ≳  k  . (2)  \nx∈Σnk , ∗ m  \nSee [16, Theorem 2] and [1, Theorem 6] . Throughout this paper, we write T1 = Ω(T2 ) or T1 ≳ T2 to denote T1 ≥ cT2 for some universal constant c > 0.  \nUnder Gaussian design A that has i.i.d. N(0, 1) entries, it was shown [22] that an efficient algorithm called normalized binary iterative hard thresholding (NBIHT) achieves the lower bound (2) up to logarithmic factors, i.e. ,  \nxpnk, ∗ ∥ˆxnbiht − x∥2 =  􀀐 km􀀑 , (3)  \nwhere ˆxnbiht denotes the output of NBIHT. (See also the earlier analysis in [14].) By convention, T1anwe( lod) ohlrieTdette1lrTaror2 dithveenmctiotcorefasscahodatrsmTofat1r(i,Cns, T)r2iengfoOulras(ro)lmaetudersniΩfv(oer)r,saresclsalcpaoernctsstantively. CT>hr0ou, anghdouhuesepae(·r), However, signals in real applications are typically only approximately sparse rather than exactly living in Σnk for some k. For any q > 0, let Bnq = {u ∈ Rn : ∥u∥q := (P |ui|q )1/q ≤ 1} denote the unit ℓq ball. The most common formulation of the set of approximately (or effectively) k-sparse  \nsignals in the unit Euclidean ball Bn2 is via  \nK 1,k := √kBn1 ∩ Bn2 .  \nThis is a relaxation of Σnk ∩ Bn2 in view of K 1,k ⊃ Σnk ∩ Bn2, and indeed, K 1,k can be viewed as the convex hull of Σnk ∩Bn2 [26, Lemma 3.1] . In the sequel, we may refer to the vectors in K 1,k as ℓ1-sparse signals, implicitly for some sparsity level k.  \nMotivated by this consideration, previous works have established a number of recovery guarantees for 1bCS of x ∈ K ∗1,k := √kBn1 ∩ Sn−1 [25 , 26 , 3 , 8 , 6 , 9] . For simplicity we assume k is known. ","cbCaiayCvboNxE9B","https://ap.wps.com/l/cbCaiayCvboNxE9B","pdf",667502,1,25,"English","en",105,"# Abstract\n# Introduction\n## One-bit compressed sensing formulation\n## Approximately sparse signal models\n## Prior recovery guarantees and algorithms\n## Dithering model and motivation for lower bounds","[{\"question\":\"What problem does the paper study in one-bit compressed sensing?\",\"answer\":\"The paper studies information-theoretic lower bounds for recovering approximately sparse signals from the signs of linear measurements in the one-bit compressed sensing model.\"},{\"question\":\"How does the paper model approximately sparse signals?\",\"answer\":\"It uses a scaled ℓ1-ball relaxation of exact k-sparsity, and extends the analysis to signals in properly scaled ℓq balls for q in [0,1].\"},{\"question\":\"What is the key strategy used to prove the lower bounds?\",\"answer\":\"The proof constructs two signals inside a small Euclidean ball that are separated in ℓ2 distance (up to logarithmic factors) yet become indistinguishable under the binary measurement outcomes.\"}]",1784199620,63,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"near-optimal-lower-bounds-on-one-bit-compressed-sensing-of-approximately-sparse-signals","",{"@graph":35,"@context":85},[36,53,68],{"@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/near-optimal-lower-bounds-on-one-bit-compressed-sensing-of-approximately-sparse-signals/84943/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper study in one-bit compressed sensing?","Question",{"text":75,"@type":76},"The paper studies information-theoretic lower bounds for recovering approximately sparse signals from the signs of linear measurements in the one-bit compressed sensing model.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper model approximately sparse signals?",{"text":80,"@type":76},"It uses a scaled ℓ1-ball relaxation of exact k-sparsity, and extends the analysis to signals in properly scaled ℓq balls for q in [0,1].",{"name":82,"@type":73,"acceptedAnswer":83},"What is the key strategy used to prove the lower bounds?",{"text":84,"@type":76},"The proof constructs two signals inside a small Euclidean ball that are separated in ℓ2 distance (up to logarithmic factors) yet become indistinguishable under the binary measurement outcomes.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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