Scaling positive random matrices: concentration and asymptotic convergence Article Swipe
It is well known that any positive matrix can be scaled to have prescribed\nrow and column sums by multiplying its rows and columns by certain positive\nscaling factors (which are unique up to a positive scalar). This procedure is\nknown as matrix scaling, and has found numerous applications in operations\nresearch, economics, image processing, and machine learning. In this work, we\ninvestigate the behavior of the scaling factors and the resulting scaled matrix\nwhen the matrix to be scaled is random. Specifically, letting\n$\\widetilde{A}\\in\\mathbb{R}^{M\\times N}$ be a positive and bounded random\nmatrix whose entries assume a certain type of independence, we provide a\nconcentration inequality for the scaling factors of $\\widetilde{A}$ around\nthose of $A = \\mathbb{E}[\\widetilde{A}]$. This result is employed to bound the\nconvergence rate of the scaling factors of $\\widetilde{A}$ to those of $A$, as\nwell as the concentration of the scaled version of $\\widetilde{A}$ around the\nscaled version of $A$ in operator norm, as $M,N\\rightarrow\\infty$. When the\nentries of $\\widetilde{A}$ are independent, $M=N$, and all prescribed row and\ncolumn sums are $1$ (i.e., doubly-stochastic matrix scaling), both of the\npreviously-mentioned bounds are $\\mathcal{O}(\\sqrt{\\log N / N})$ with high\nprobability. We demonstrate our results in several simulations.\n
Related Topics
Concepts
Scaling
Mathematics
Combinatorics
Matrix (chemical analysis)
Random matrix
Rate of convergence
Bounded function
Scalar (mathematics)
Concentration inequality
Discrete mathematics
Mathematical analysis
Eigenvalues and eigenvectors
Geometry
Physics
Computer science
Composite material
Materials science
Quantum mechanics
Channel (broadcasting)
Computer network
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1214/22-ecp502
- https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdf
- OA Status
- gold
- Cited By
- 2
- References
- 39
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W3111170930
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W3111170930Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1214/22-ecp502Digital Object Identifier
- Title
-
Scaling positive random matrices: concentration and asymptotic convergenceWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-01-01Full publication date if available
- Authors
-
Boris LandaList of authors in order
- Landing page
-
https://doi.org/10.1214/22-ecp502Publisher landing page
- PDF URL
-
https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdfDirect link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
goldOpen access status per OpenAlex
- OA URL
-
https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdfDirect OA link when available
- Concepts
-
Scaling, Mathematics, Combinatorics, Matrix (chemical analysis), Random matrix, Rate of convergence, Bounded function, Scalar (mathematics), Concentration inequality, Discrete mathematics, Mathematical analysis, Eigenvalues and eigenvectors, Geometry, Physics, Computer science, Composite material, Materials science, Quantum mechanics, Channel (broadcasting), Computer networkTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
2Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 1, 2022: 1Per-year citation counts (last 5 years)
- References (count)
-
39Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W3111170930 |
|---|---|
| doi | https://doi.org/10.1214/22-ecp502 |
| ids.doi | https://doi.org/10.1214/22-ecp502 |
| ids.mag | 3111170930 |
| ids.openalex | https://openalex.org/W3111170930 |
| fwci | 0.83488166 |
| type | article |
| title | Scaling positive random matrices: concentration and asymptotic convergence |
| biblio.issue | none |
| biblio.volume | 27 |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T11716 |
| topics[0].field.id | https://openalex.org/fields/26 |
| topics[0].field.display_name | Mathematics |
| topics[0].score | 0.9955000281333923 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2613 |
| topics[0].subfield.display_name | Statistics and Probability |
| topics[0].display_name | Random Matrices and Applications |
| topics[1].id | https://openalex.org/T11830 |
| topics[1].field.id | https://openalex.org/fields/26 |
| topics[1].field.display_name | Mathematics |
| topics[1].score | 0.9955000281333923 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/2604 |
| topics[1].subfield.display_name | Applied Mathematics |
| topics[1].display_name | Point processes and geometric inequalities |
| topics[2].id | https://openalex.org/T12056 |
| topics[2].field.id | https://openalex.org/fields/26 |
| topics[2].field.display_name | Mathematics |
| topics[2].score | 0.9825999736785889 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/2613 |
| topics[2].subfield.display_name | Statistics and Probability |
| topics[2].display_name | Markov Chains and Monte Carlo Methods |
| is_xpac | False |
| apc_list.value | 675 |
| apc_list.currency | USD |
| apc_list.value_usd | 675 |
| apc_paid.value | 675 |
| apc_paid.currency | USD |
| apc_paid.value_usd | 675 |
| concepts[0].id | https://openalex.org/C99844830 |
| concepts[0].level | 2 |
| concepts[0].score | 0.8107335567474365 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q102441924 |
| concepts[0].display_name | Scaling |
| concepts[1].id | https://openalex.org/C33923547 |
| concepts[1].level | 0 |
| concepts[1].score | 0.7556905150413513 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[1].display_name | Mathematics |
| concepts[2].id | https://openalex.org/C114614502 |
| concepts[2].level | 1 |
| concepts[2].score | 0.5925682783126831 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q76592 |
| concepts[2].display_name | Combinatorics |
| concepts[3].id | https://openalex.org/C106487976 |
| concepts[3].level | 2 |
| concepts[3].score | 0.5639936327934265 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q685816 |
| concepts[3].display_name | Matrix (chemical analysis) |
| concepts[4].id | https://openalex.org/C64812099 |
| concepts[4].level | 3 |
| concepts[4].score | 0.5467076301574707 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q176604 |
| concepts[4].display_name | Random matrix |
| concepts[5].id | https://openalex.org/C57869625 |
| concepts[5].level | 3 |
| concepts[5].score | 0.5225031971931458 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q1783502 |
| concepts[5].display_name | Rate of convergence |
| concepts[6].id | https://openalex.org/C34388435 |
| concepts[6].level | 2 |
| concepts[6].score | 0.5118941068649292 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q2267362 |
| concepts[6].display_name | Bounded function |
| concepts[7].id | https://openalex.org/C57691317 |
| concepts[7].level | 2 |
| concepts[7].score | 0.451949805021286 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q1289248 |
| concepts[7].display_name | Scalar (mathematics) |
| concepts[8].id | https://openalex.org/C2777169571 |
| concepts[8].level | 2 |
| concepts[8].score | 0.429656982421875 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q5158328 |
| concepts[8].display_name | Concentration inequality |
| concepts[9].id | https://openalex.org/C118615104 |
| concepts[9].level | 1 |
| concepts[9].score | 0.4030502438545227 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q121416 |
| concepts[9].display_name | Discrete mathematics |
| concepts[10].id | https://openalex.org/C134306372 |
| concepts[10].level | 1 |
| concepts[10].score | 0.21287992596626282 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[10].display_name | Mathematical analysis |
| concepts[11].id | https://openalex.org/C158693339 |
| concepts[11].level | 2 |
| concepts[11].score | 0.11640793085098267 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q190524 |
| concepts[11].display_name | Eigenvalues and eigenvectors |
| concepts[12].id | https://openalex.org/C2524010 |
| concepts[12].level | 1 |
| concepts[12].score | 0.10831373929977417 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q8087 |
| concepts[12].display_name | Geometry |
| concepts[13].id | https://openalex.org/C121332964 |
| concepts[13].level | 0 |
| concepts[13].score | 0.07682698965072632 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[13].display_name | Physics |
| concepts[14].id | https://openalex.org/C41008148 |
| concepts[14].level | 0 |
| concepts[14].score | 0.06859320402145386 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[14].display_name | Computer science |
| concepts[15].id | https://openalex.org/C159985019 |
| concepts[15].level | 1 |
| concepts[15].score | 0.0 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q181790 |
| concepts[15].display_name | Composite material |
| concepts[16].id | https://openalex.org/C192562407 |
| concepts[16].level | 0 |
| concepts[16].score | 0.0 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q228736 |
| concepts[16].display_name | Materials science |
| concepts[17].id | https://openalex.org/C62520636 |
| concepts[17].level | 1 |
| concepts[17].score | 0.0 |
| concepts[17].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[17].display_name | Quantum mechanics |
| concepts[18].id | https://openalex.org/C127162648 |
| concepts[18].level | 2 |
| concepts[18].score | 0.0 |
| concepts[18].wikidata | https://www.wikidata.org/wiki/Q16858953 |
| concepts[18].display_name | Channel (broadcasting) |
| concepts[19].id | https://openalex.org/C31258907 |
| concepts[19].level | 1 |
| concepts[19].score | 0.0 |
| concepts[19].wikidata | https://www.wikidata.org/wiki/Q1301371 |
| concepts[19].display_name | Computer network |
| keywords[0].id | https://openalex.org/keywords/scaling |
| keywords[0].score | 0.8107335567474365 |
| keywords[0].display_name | Scaling |
| keywords[1].id | https://openalex.org/keywords/mathematics |
| keywords[1].score | 0.7556905150413513 |
| keywords[1].display_name | Mathematics |
| keywords[2].id | https://openalex.org/keywords/combinatorics |
| keywords[2].score | 0.5925682783126831 |
| keywords[2].display_name | Combinatorics |
| keywords[3].id | https://openalex.org/keywords/matrix |
| keywords[3].score | 0.5639936327934265 |
| keywords[3].display_name | Matrix (chemical analysis) |
| keywords[4].id | https://openalex.org/keywords/random-matrix |
| keywords[4].score | 0.5467076301574707 |
| keywords[4].display_name | Random matrix |
| keywords[5].id | https://openalex.org/keywords/rate-of-convergence |
| keywords[5].score | 0.5225031971931458 |
| keywords[5].display_name | Rate of convergence |
| keywords[6].id | https://openalex.org/keywords/bounded-function |
| keywords[6].score | 0.5118941068649292 |
| keywords[6].display_name | Bounded function |
| keywords[7].id | https://openalex.org/keywords/scalar |
| keywords[7].score | 0.451949805021286 |
| keywords[7].display_name | Scalar (mathematics) |
| keywords[8].id | https://openalex.org/keywords/concentration-inequality |
| keywords[8].score | 0.429656982421875 |
| keywords[8].display_name | Concentration inequality |
| keywords[9].id | https://openalex.org/keywords/discrete-mathematics |
| keywords[9].score | 0.4030502438545227 |
| keywords[9].display_name | Discrete mathematics |
| keywords[10].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[10].score | 0.21287992596626282 |
| keywords[10].display_name | Mathematical analysis |
| keywords[11].id | https://openalex.org/keywords/eigenvalues-and-eigenvectors |
| keywords[11].score | 0.11640793085098267 |
| keywords[11].display_name | Eigenvalues and eigenvectors |
| keywords[12].id | https://openalex.org/keywords/geometry |
| keywords[12].score | 0.10831373929977417 |
| keywords[12].display_name | Geometry |
| keywords[13].id | https://openalex.org/keywords/physics |
| keywords[13].score | 0.07682698965072632 |
| keywords[13].display_name | Physics |
| keywords[14].id | https://openalex.org/keywords/computer-science |
| keywords[14].score | 0.06859320402145386 |
| keywords[14].display_name | Computer science |
| language | en |
| locations[0].id | doi:10.1214/22-ecp502 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S87494164 |
| locations[0].source.issn | 1083-589X |
| locations[0].source.type | journal |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | 1083-589X |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | True |
| locations[0].source.display_name | Electronic Communications in Probability |
| locations[0].source.host_organization | https://openalex.org/P4310319881 |
| locations[0].source.host_organization_name | Institute of Mathematical Statistics |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310319881 |
| locations[0].license | cc-by |
| locations[0].pdf_url | https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdf |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Electronic Communications in Probability |
| locations[0].landing_page_url | https://doi.org/10.1214/22-ecp502 |
| locations[1].id | pmh:oai:arXiv.org:2012.06393 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | True |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | arXiv (Cornell University) |
| locations[1].source.host_organization | https://openalex.org/I205783295 |
| locations[1].source.host_organization_name | Cornell University |
| locations[1].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[1].license | |
| locations[1].pdf_url | https://arxiv.org/pdf/2012.06393 |
| locations[1].version | submittedVersion |
| locations[1].raw_type | text |
| locations[1].license_id | |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | http://arxiv.org/abs/2012.06393 |
| indexed_in | arxiv, crossref, doaj |
| authorships[0].author.id | https://openalex.org/A5031605523 |
| authorships[0].author.orcid | |
| authorships[0].author.display_name | Boris Landa |
| authorships[0].countries | US |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I32971472 |
| authorships[0].affiliations[0].raw_affiliation_string | Department of Mathematics, Yale University |
| authorships[0].institutions[0].id | https://openalex.org/I32971472 |
| authorships[0].institutions[0].ror | https://ror.org/03v76x132 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I32971472 |
| authorships[0].institutions[0].country_code | US |
| authorships[0].institutions[0].display_name | Yale University |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Boris Landa |
| authorships[0].is_corresponding | True |
| authorships[0].raw_affiliation_strings | Department of Mathematics, Yale University |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdf |
| open_access.oa_status | gold |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Scaling positive random matrices: concentration and asymptotic convergence |
| has_fulltext | True |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T11716 |
| primary_topic.field.id | https://openalex.org/fields/26 |
| primary_topic.field.display_name | Mathematics |
| primary_topic.score | 0.9955000281333923 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2613 |
| primary_topic.subfield.display_name | Statistics and Probability |
| primary_topic.display_name | Random Matrices and Applications |
| related_works | https://openalex.org/W4378770497, https://openalex.org/W2049584446, https://openalex.org/W2079781215, https://openalex.org/W2064404759, https://openalex.org/W4385571583, https://openalex.org/W4308245303, https://openalex.org/W2014033564, https://openalex.org/W2910573937, https://openalex.org/W4322825348, https://openalex.org/W3194228126 |
| cited_by_count | 2 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 1 |
| counts_by_year[1].year | 2022 |
| counts_by_year[1].cited_by_count | 1 |
| locations_count | 2 |
| best_oa_location.id | doi:10.1214/22-ecp502 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S87494164 |
| best_oa_location.source.issn | 1083-589X |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | 1083-589X |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | True |
| best_oa_location.source.display_name | Electronic Communications in Probability |
| best_oa_location.source.host_organization | https://openalex.org/P4310319881 |
| best_oa_location.source.host_organization_name | Institute of Mathematical Statistics |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310319881 |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdf |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Electronic Communications in Probability |
| best_oa_location.landing_page_url | https://doi.org/10.1214/22-ecp502 |
| primary_location.id | doi:10.1214/22-ecp502 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S87494164 |
| primary_location.source.issn | 1083-589X |
| primary_location.source.type | journal |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | 1083-589X |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | True |
| primary_location.source.display_name | Electronic Communications in Probability |
| primary_location.source.host_organization | https://openalex.org/P4310319881 |
| primary_location.source.host_organization_name | Institute of Mathematical Statistics |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310319881 |
| primary_location.license | cc-by |
| primary_location.pdf_url | https://projecteuclid.org/journals/electronic-communications-in-probability/volume-27/issue-none/Scaling-positive-random-matrices-concentration-and-asymptotic-convergence/10.1214/22-ECP502.pdf |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Electronic Communications in Probability |
| primary_location.landing_page_url | https://doi.org/10.1214/22-ecp502 |
| publication_date | 2022-01-01 |
| publication_year | 2022 |
| referenced_works | https://openalex.org/W3175252931, https://openalex.org/W2913699596, https://openalex.org/W2092040882, https://openalex.org/W1975457501, https://openalex.org/W2963830216, https://openalex.org/W1587878450, https://openalex.org/W2158256890, https://openalex.org/W6635085931, https://openalex.org/W1977842112, https://openalex.org/W3168712283, https://openalex.org/W2103012681, https://openalex.org/W3033420020, https://openalex.org/W2079639858, https://openalex.org/W2073219008, https://openalex.org/W3137945549, https://openalex.org/W2060220641, https://openalex.org/W2964095333, https://openalex.org/W1963816084, https://openalex.org/W3217740066, https://openalex.org/W4206471589, https://openalex.org/W2011848437, https://openalex.org/W1990283121, https://openalex.org/W2312355711, https://openalex.org/W2003447360, https://openalex.org/W2051267445, https://openalex.org/W4250954493, https://openalex.org/W3175639804, https://openalex.org/W2161997479, https://openalex.org/W3108682107, https://openalex.org/W2158131535, https://openalex.org/W2787248994, https://openalex.org/W2088706743, https://openalex.org/W3033631085, https://openalex.org/W1980494867, https://openalex.org/W2951798229, https://openalex.org/W2522454188, https://openalex.org/W3101974274, https://openalex.org/W2963508016, https://openalex.org/W3039256091 |
| referenced_works_count | 39 |
| abstract_inverted_index./ | 172 |
| abstract_inverted_index.= | 106 |
| abstract_inverted_index.N | 171 |
| abstract_inverted_index.a | 32, 80, 88 |
| abstract_inverted_index.$A | 105 |
| abstract_inverted_index.In | 54 |
| abstract_inverted_index.It | 0 |
| abstract_inverted_index.We | 176 |
| abstract_inverted_index.as | 38, 127, 144 |
| abstract_inverted_index.be | 9, 72, 79 |
| abstract_inverted_index.by | 17, 23 |
| abstract_inverted_index.in | 46, 141, 180 |
| abstract_inverted_index.is | 1, 74, 110 |
| abstract_inverted_index.of | 60, 91, 101, 104, 116, 120, 124, 130, 134, 139, 148, 166 |
| abstract_inverted_index.to | 11, 31, 71, 112, 122 |
| abstract_inverted_index.up | 30 |
| abstract_inverted_index.we | 93 |
| abstract_inverted_index.$1$ | 160 |
| abstract_inverted_index.$A$ | 140 |
| abstract_inverted_index.N}$ | 78 |
| abstract_inverted_index.all | 154 |
| abstract_inverted_index.and | 14, 21, 41, 51, 64, 82, 153 |
| abstract_inverted_index.any | 5 |
| abstract_inverted_index.are | 28, 150, 159, 169 |
| abstract_inverted_index.can | 8 |
| abstract_inverted_index.for | 97 |
| abstract_inverted_index.has | 42 |
| abstract_inverted_index.its | 19 |
| abstract_inverted_index.our | 178 |
| abstract_inverted_index.row | 156 |
| abstract_inverted_index.the | 58, 61, 65, 69, 98, 117, 128, 131 |
| abstract_inverted_index.$A$, | 125 |
| abstract_inverted_index.N})$ | 173 |
| abstract_inverted_index.This | 35, 108 |
| abstract_inverted_index.When | 146 |
| abstract_inverted_index.both | 165 |
| abstract_inverted_index.have | 12 |
| abstract_inverted_index.rate | 115 |
| abstract_inverted_index.rows | 20 |
| abstract_inverted_index.sums | 16, 158 |
| abstract_inverted_index.that | 4 |
| abstract_inverted_index.this | 55 |
| abstract_inverted_index.type | 90 |
| abstract_inverted_index.well | 2 |
| abstract_inverted_index.with | 174 |
| abstract_inverted_index.bound | 113 |
| abstract_inverted_index.found | 43 |
| abstract_inverted_index.image | 49 |
| abstract_inverted_index.known | 3 |
| abstract_inverted_index.norm, | 143 |
| abstract_inverted_index.those | 123 |
| abstract_inverted_index.whose | 85 |
| abstract_inverted_index.work, | 56 |
| abstract_inverted_index.$M=N$, | 152 |
| abstract_inverted_index.(i.e., | 161 |
| abstract_inverted_index.(which | 27 |
| abstract_inverted_index.around | 136 |
| abstract_inverted_index.assume | 87 |
| abstract_inverted_index.bounds | 168 |
| abstract_inverted_index.column | 15 |
| abstract_inverted_index.matrix | 7, 39, 70, 163 |
| abstract_inverted_index.result | 109 |
| abstract_inverted_index.scaled | 10, 67, 73, 132 |
| abstract_inverted_index.unique | 29 |
| abstract_inverted_index.bounded | 83 |
| abstract_inverted_index.certain | 24, 89 |
| abstract_inverted_index.columns | 22 |
| abstract_inverted_index.entries | 86 |
| abstract_inverted_index.factors | 26, 63, 100, 119 |
| abstract_inverted_index.machine | 52 |
| abstract_inverted_index.provide | 94 |
| abstract_inverted_index.random. | 75 |
| abstract_inverted_index.results | 179 |
| abstract_inverted_index.scaling | 62, 99, 118 |
| abstract_inverted_index.several | 181 |
| abstract_inverted_index.version | 133, 138 |
| abstract_inverted_index.as\nwell | 126 |
| abstract_inverted_index.behavior | 59 |
| abstract_inverted_index.employed | 111 |
| abstract_inverted_index.numerous | 44 |
| abstract_inverted_index.operator | 142 |
| abstract_inverted_index.positive | 6, 33, 81 |
| abstract_inverted_index.scalar). | 34 |
| abstract_inverted_index.scaling, | 40 |
| abstract_inverted_index.is\nknown | 37 |
| abstract_inverted_index.learning. | 53 |
| abstract_inverted_index.procedure | 36 |
| abstract_inverted_index.resulting | 66 |
| abstract_inverted_index.scaling), | 164 |
| abstract_inverted_index.economics, | 48 |
| abstract_inverted_index.inequality | 96 |
| abstract_inverted_index.prescribed | 155 |
| abstract_inverted_index.and\ncolumn | 157 |
| abstract_inverted_index.demonstrate | 177 |
| abstract_inverted_index.multiplying | 18 |
| abstract_inverted_index.processing, | 50 |
| abstract_inverted_index.the\nscaled | 137 |
| abstract_inverted_index.applications | 45 |
| abstract_inverted_index.independent, | 151 |
| abstract_inverted_index.matrix\nwhen | 68 |
| abstract_inverted_index.the\nentries | 147 |
| abstract_inverted_index.Specifically, | 76 |
| abstract_inverted_index.around\nthose | 103 |
| abstract_inverted_index.concentration | 129 |
| abstract_inverted_index.independence, | 92 |
| abstract_inverted_index.random\nmatrix | 84 |
| abstract_inverted_index.simulations.\n | 182 |
| abstract_inverted_index.prescribed\nrow | 13 |
| abstract_inverted_index.we\ninvestigate | 57 |
| abstract_inverted_index.$\\widetilde{A}$ | 102, 121, 135, 149 |
| abstract_inverted_index.a\nconcentration | 95 |
| abstract_inverted_index.the\nconvergence | 114 |
| abstract_inverted_index.doubly-stochastic | 162 |
| abstract_inverted_index.positive\nscaling | 25 |
| abstract_inverted_index.high\nprobability. | 175 |
| abstract_inverted_index.operations\nresearch, | 47 |
| abstract_inverted_index.$M,N\\rightarrow\\infty$. | 145 |
| abstract_inverted_index.the\npreviously-mentioned | 167 |
| abstract_inverted_index.$\\mathcal{O}(\\sqrt{\\log | 170 |
| abstract_inverted_index.\\mathbb{E}[\\widetilde{A}]$. | 107 |
| abstract_inverted_index.letting\n$\\widetilde{A}\\in\\mathbb{R}^{M\\times | 77 |
| cited_by_percentile_year.max | 95 |
| cited_by_percentile_year.min | 89 |
| corresponding_author_ids | https://openalex.org/A5031605523 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 1 |
| corresponding_institution_ids | https://openalex.org/I32971472 |
| citation_normalized_percentile.value | 0.62884185 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |