On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds Article Swipe
Ming Jin
,
Vanshaj Khattar
,
Harshal D. Kaushik
,
Bilgehan Sel
,
Ruoxi Jia
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1609/aaai.v37i7.25981
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1609/aaai.v37i7.25981
We study the expressibility and learnability of solution functions of convex optimization and their multi-layer architectural extension. The main results are: (1) the class of solution functions of linear programming (LP) and quadratic programming (QP) is a universal approximant for the smooth model class or some restricted Sobolev space, and we characterize the rate-distortion, (2) the approximation power is investigated through a viewpoint of regression error, where information about the target function is provided in terms of data observations, (3) compositionality in the form of deep architecture with optimization as a layer is shown to reconstruct some basic functions used in numerical analysis without error, which implies that (4) a substantial reduction in rate-distortion can be achieved with a universal network architecture, and (5) we discuss the statistical bounds of empirical covering numbers for LP/QP, as well as a generic optimization problem (possibly nonconvex) by exploiting tame geometry. Our results provide the **first rigorous analysis of the approximation and learning-theoretic properties of solution functions** with implications for algorithmic design and performance guarantees.
Related Topics
Concepts
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1609/aaai.v37i7.25981
- https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753
- OA Status
- diamond
- Cited By
- 7
- References
- 81
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4382467878
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4382467878Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1609/aaai.v37i7.25981Digital Object Identifier
- Title
-
On Solution Functions of Optimization: Universal Approximation and Covering Number BoundsWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2023Year of publication
- Publication date
-
2023-06-26Full publication date if available
- Authors
-
Ming Jin, Vanshaj Khattar, Harshal D. Kaushik, Bilgehan Sel, Ruoxi JiaList of authors in order
- Landing page
-
https://doi.org/10.1609/aaai.v37i7.25981Publisher landing page
- PDF URL
-
https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
diamondOpen access status per OpenAlex
- OA URL
-
https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753Direct OA link when available
- Concepts
-
Sobolev space, Mathematics, Approximation error, Quadratic programming, Mathematical optimization, Linear programming, Optimization problem, Computer science, Applied mathematics, Algorithm, Discrete mathematics, Pure mathematicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
7Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 2, 2024: 2, 2023: 3Per-year citation counts (last 5 years)
- References (count)
-
81Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4382467878 |
|---|---|
| doi | https://doi.org/10.1609/aaai.v37i7.25981 |
| ids.doi | https://doi.org/10.1609/aaai.v37i7.25981 |
| ids.openalex | https://openalex.org/W4382467878 |
| fwci | 3.71308289 |
| type | article |
| title | On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds |
| biblio.issue | 7 |
| biblio.volume | 37 |
| biblio.last_page | 8131 |
| biblio.first_page | 8123 |
| topics[0].id | https://openalex.org/T10500 |
| topics[0].field.id | https://openalex.org/fields/22 |
| topics[0].field.display_name | Engineering |
| topics[0].score | 0.9973999857902527 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2206 |
| topics[0].subfield.display_name | Computational Mechanics |
| topics[0].display_name | Sparse and Compressive Sensing Techniques |
| topics[1].id | https://openalex.org/T12072 |
| topics[1].field.id | https://openalex.org/fields/17 |
| topics[1].field.display_name | Computer Science |
| topics[1].score | 0.9886000156402588 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1702 |
| topics[1].subfield.display_name | Artificial Intelligence |
| topics[1].display_name | Machine Learning and Algorithms |
| topics[2].id | https://openalex.org/T10136 |
| topics[2].field.id | https://openalex.org/fields/26 |
| topics[2].field.display_name | Mathematics |
| topics[2].score | 0.9847000241279602 |
| 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 | Statistical Methods and Inference |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C99730327 |
| concepts[0].level | 2 |
| concepts[0].score | 0.5707512497901917 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q1501536 |
| concepts[0].display_name | Sobolev space |
| concepts[1].id | https://openalex.org/C33923547 |
| concepts[1].level | 0 |
| concepts[1].score | 0.5608202219009399 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[1].display_name | Mathematics |
| concepts[2].id | https://openalex.org/C122383733 |
| concepts[2].level | 2 |
| concepts[2].score | 0.5019721984863281 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q865920 |
| concepts[2].display_name | Approximation error |
| concepts[3].id | https://openalex.org/C81845259 |
| concepts[3].level | 2 |
| concepts[3].score | 0.45288705825805664 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q290117 |
| concepts[3].display_name | Quadratic programming |
| concepts[4].id | https://openalex.org/C126255220 |
| concepts[4].level | 1 |
| concepts[4].score | 0.4437357783317566 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q141495 |
| concepts[4].display_name | Mathematical optimization |
| concepts[5].id | https://openalex.org/C41045048 |
| concepts[5].level | 2 |
| concepts[5].score | 0.42608463764190674 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q202843 |
| concepts[5].display_name | Linear programming |
| concepts[6].id | https://openalex.org/C137836250 |
| concepts[6].level | 2 |
| concepts[6].score | 0.4130958318710327 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q984063 |
| concepts[6].display_name | Optimization problem |
| concepts[7].id | https://openalex.org/C41008148 |
| concepts[7].level | 0 |
| concepts[7].score | 0.3759523332118988 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[7].display_name | Computer science |
| concepts[8].id | https://openalex.org/C28826006 |
| concepts[8].level | 1 |
| concepts[8].score | 0.37566643953323364 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q33521 |
| concepts[8].display_name | Applied mathematics |
| concepts[9].id | https://openalex.org/C11413529 |
| concepts[9].level | 1 |
| concepts[9].score | 0.3409348726272583 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q8366 |
| concepts[9].display_name | Algorithm |
| concepts[10].id | https://openalex.org/C118615104 |
| concepts[10].level | 1 |
| concepts[10].score | 0.32815465331077576 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q121416 |
| concepts[10].display_name | Discrete mathematics |
| concepts[11].id | https://openalex.org/C202444582 |
| concepts[11].level | 1 |
| concepts[11].score | 0.21955248713493347 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q837863 |
| concepts[11].display_name | Pure mathematics |
| keywords[0].id | https://openalex.org/keywords/sobolev-space |
| keywords[0].score | 0.5707512497901917 |
| keywords[0].display_name | Sobolev space |
| keywords[1].id | https://openalex.org/keywords/mathematics |
| keywords[1].score | 0.5608202219009399 |
| keywords[1].display_name | Mathematics |
| keywords[2].id | https://openalex.org/keywords/approximation-error |
| keywords[2].score | 0.5019721984863281 |
| keywords[2].display_name | Approximation error |
| keywords[3].id | https://openalex.org/keywords/quadratic-programming |
| keywords[3].score | 0.45288705825805664 |
| keywords[3].display_name | Quadratic programming |
| keywords[4].id | https://openalex.org/keywords/mathematical-optimization |
| keywords[4].score | 0.4437357783317566 |
| keywords[4].display_name | Mathematical optimization |
| keywords[5].id | https://openalex.org/keywords/linear-programming |
| keywords[5].score | 0.42608463764190674 |
| keywords[5].display_name | Linear programming |
| keywords[6].id | https://openalex.org/keywords/optimization-problem |
| keywords[6].score | 0.4130958318710327 |
| keywords[6].display_name | Optimization problem |
| keywords[7].id | https://openalex.org/keywords/computer-science |
| keywords[7].score | 0.3759523332118988 |
| keywords[7].display_name | Computer science |
| keywords[8].id | https://openalex.org/keywords/applied-mathematics |
| keywords[8].score | 0.37566643953323364 |
| keywords[8].display_name | Applied mathematics |
| keywords[9].id | https://openalex.org/keywords/algorithm |
| keywords[9].score | 0.3409348726272583 |
| keywords[9].display_name | Algorithm |
| keywords[10].id | https://openalex.org/keywords/discrete-mathematics |
| keywords[10].score | 0.32815465331077576 |
| keywords[10].display_name | Discrete mathematics |
| keywords[11].id | https://openalex.org/keywords/pure-mathematics |
| keywords[11].score | 0.21955248713493347 |
| keywords[11].display_name | Pure mathematics |
| language | en |
| locations[0].id | doi:10.1609/aaai.v37i7.25981 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4210191458 |
| locations[0].source.issn | 2159-5399, 2374-3468 |
| locations[0].source.type | conference |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | 2159-5399 |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Proceedings of the AAAI Conference on Artificial Intelligence |
| locations[0].source.host_organization | https://openalex.org/P4310320058 |
| locations[0].source.host_organization_name | Association for the Advancement of Artificial Intelligence |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310320058 |
| locations[0].source.host_organization_lineage_names | Association for the Advancement of Artificial Intelligence |
| locations[0].license | |
| locations[0].pdf_url | https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753 |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Proceedings of the AAAI Conference on Artificial Intelligence |
| locations[0].landing_page_url | https://doi.org/10.1609/aaai.v37i7.25981 |
| indexed_in | crossref |
| authorships[0].author.id | https://openalex.org/A5012672471 |
| authorships[0].author.orcid | https://orcid.org/0000-0002-9824-7647 |
| authorships[0].author.display_name | Ming Jin |
| authorships[0].countries | US |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I859038795 |
| authorships[0].affiliations[0].raw_affiliation_string | Virginia Tech |
| authorships[0].institutions[0].id | https://openalex.org/I859038795 |
| authorships[0].institutions[0].ror | https://ror.org/02smfhw86 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I859038795 |
| authorships[0].institutions[0].country_code | US |
| authorships[0].institutions[0].display_name | Virginia Tech |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Ming Jin |
| authorships[0].is_corresponding | False |
| authorships[0].raw_affiliation_strings | Virginia Tech |
| authorships[1].author.id | https://openalex.org/A5070497653 |
| authorships[1].author.orcid | https://orcid.org/0009-0006-0734-5804 |
| authorships[1].author.display_name | Vanshaj Khattar |
| authorships[1].countries | US |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I859038795 |
| authorships[1].affiliations[0].raw_affiliation_string | Virginia Tech |
| authorships[1].institutions[0].id | https://openalex.org/I859038795 |
| authorships[1].institutions[0].ror | https://ror.org/02smfhw86 |
| authorships[1].institutions[0].type | education |
| authorships[1].institutions[0].lineage | https://openalex.org/I859038795 |
| authorships[1].institutions[0].country_code | US |
| authorships[1].institutions[0].display_name | Virginia Tech |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Vanshaj Khattar |
| authorships[1].is_corresponding | False |
| authorships[1].raw_affiliation_strings | Virginia Tech |
| authorships[2].author.id | https://openalex.org/A5020457099 |
| authorships[2].author.orcid | |
| authorships[2].author.display_name | Harshal D. Kaushik |
| authorships[2].countries | US |
| authorships[2].affiliations[0].institution_ids | https://openalex.org/I859038795 |
| authorships[2].affiliations[0].raw_affiliation_string | Virginia Tech |
| authorships[2].institutions[0].id | https://openalex.org/I859038795 |
| authorships[2].institutions[0].ror | https://ror.org/02smfhw86 |
| authorships[2].institutions[0].type | education |
| authorships[2].institutions[0].lineage | https://openalex.org/I859038795 |
| authorships[2].institutions[0].country_code | US |
| authorships[2].institutions[0].display_name | Virginia Tech |
| authorships[2].author_position | middle |
| authorships[2].raw_author_name | Harshal Kaushik |
| authorships[2].is_corresponding | False |
| authorships[2].raw_affiliation_strings | Virginia Tech |
| authorships[3].author.id | https://openalex.org/A5017539185 |
| authorships[3].author.orcid | https://orcid.org/0000-0001-8701-6539 |
| authorships[3].author.display_name | Bilgehan Sel |
| authorships[3].countries | US |
| authorships[3].affiliations[0].institution_ids | https://openalex.org/I859038795 |
| authorships[3].affiliations[0].raw_affiliation_string | Virginia Tech |
| authorships[3].institutions[0].id | https://openalex.org/I859038795 |
| authorships[3].institutions[0].ror | https://ror.org/02smfhw86 |
| authorships[3].institutions[0].type | education |
| authorships[3].institutions[0].lineage | https://openalex.org/I859038795 |
| authorships[3].institutions[0].country_code | US |
| authorships[3].institutions[0].display_name | Virginia Tech |
| authorships[3].author_position | middle |
| authorships[3].raw_author_name | Bilgehan Sel |
| authorships[3].is_corresponding | False |
| authorships[3].raw_affiliation_strings | Virginia Tech |
| authorships[4].author.id | https://openalex.org/A5032275274 |
| authorships[4].author.orcid | https://orcid.org/0000-0001-9662-9556 |
| authorships[4].author.display_name | Ruoxi Jia |
| authorships[4].countries | US |
| authorships[4].affiliations[0].institution_ids | https://openalex.org/I859038795 |
| authorships[4].affiliations[0].raw_affiliation_string | Virginia Tech |
| authorships[4].institutions[0].id | https://openalex.org/I859038795 |
| authorships[4].institutions[0].ror | https://ror.org/02smfhw86 |
| authorships[4].institutions[0].type | education |
| authorships[4].institutions[0].lineage | https://openalex.org/I859038795 |
| authorships[4].institutions[0].country_code | US |
| authorships[4].institutions[0].display_name | Virginia Tech |
| authorships[4].author_position | last |
| authorships[4].raw_author_name | Ruoxi Jia |
| authorships[4].is_corresponding | False |
| authorships[4].raw_affiliation_strings | Virginia Tech |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753 |
| open_access.oa_status | diamond |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds |
| has_fulltext | True |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T10500 |
| primary_topic.field.id | https://openalex.org/fields/22 |
| primary_topic.field.display_name | Engineering |
| primary_topic.score | 0.9973999857902527 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2206 |
| primary_topic.subfield.display_name | Computational Mechanics |
| primary_topic.display_name | Sparse and Compressive Sensing Techniques |
| related_works | https://openalex.org/W4299978068, https://openalex.org/W2950745223, https://openalex.org/W2164189677, https://openalex.org/W4301407218, https://openalex.org/W4388311156, https://openalex.org/W2895393216, https://openalex.org/W158460010, https://openalex.org/W2322281151, https://openalex.org/W2011094784, https://openalex.org/W1703868599 |
| cited_by_count | 7 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 2 |
| counts_by_year[1].year | 2024 |
| counts_by_year[1].cited_by_count | 2 |
| counts_by_year[2].year | 2023 |
| counts_by_year[2].cited_by_count | 3 |
| locations_count | 1 |
| best_oa_location.id | doi:10.1609/aaai.v37i7.25981 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4210191458 |
| best_oa_location.source.issn | 2159-5399, 2374-3468 |
| best_oa_location.source.type | conference |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | 2159-5399 |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Proceedings of the AAAI Conference on Artificial Intelligence |
| best_oa_location.source.host_organization | https://openalex.org/P4310320058 |
| best_oa_location.source.host_organization_name | Association for the Advancement of Artificial Intelligence |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310320058 |
| best_oa_location.source.host_organization_lineage_names | Association for the Advancement of Artificial Intelligence |
| best_oa_location.license | |
| best_oa_location.pdf_url | https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753 |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Proceedings of the AAAI Conference on Artificial Intelligence |
| best_oa_location.landing_page_url | https://doi.org/10.1609/aaai.v37i7.25981 |
| primary_location.id | doi:10.1609/aaai.v37i7.25981 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4210191458 |
| primary_location.source.issn | 2159-5399, 2374-3468 |
| primary_location.source.type | conference |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | 2159-5399 |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Proceedings of the AAAI Conference on Artificial Intelligence |
| primary_location.source.host_organization | https://openalex.org/P4310320058 |
| primary_location.source.host_organization_name | Association for the Advancement of Artificial Intelligence |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310320058 |
| primary_location.source.host_organization_lineage_names | Association for the Advancement of Artificial Intelligence |
| primary_location.license | |
| primary_location.pdf_url | https://ojs.aaai.org/index.php/AAAI/article/download/25981/25753 |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Proceedings of the AAAI Conference on Artificial Intelligence |
| primary_location.landing_page_url | https://doi.org/10.1609/aaai.v37i7.25981 |
| publication_date | 2023-06-26 |
| publication_year | 2023 |
| referenced_works | https://openalex.org/W3037298378, https://openalex.org/W6808467059, https://openalex.org/W6767977373, https://openalex.org/W2754641446, https://openalex.org/W1542886316, https://openalex.org/W2046981034, https://openalex.org/W2144935407, https://openalex.org/W213438331, https://openalex.org/W2800415562, https://openalex.org/W2115881141, https://openalex.org/W1971547063, https://openalex.org/W2089947415, https://openalex.org/W6655745416, https://openalex.org/W6665018949, https://openalex.org/W2965067322, https://openalex.org/W6662091678, https://openalex.org/W2798818710, https://openalex.org/W6787868878, https://openalex.org/W2278798990, https://openalex.org/W2958855579, https://openalex.org/W4237326927, https://openalex.org/W2483216001, https://openalex.org/W3099132019, https://openalex.org/W2103183297, https://openalex.org/W6785483306, https://openalex.org/W1969280568, https://openalex.org/W2992833799, https://openalex.org/W6670182552, https://openalex.org/W2137983211, https://openalex.org/W6656668297, https://openalex.org/W2885456264, https://openalex.org/W4226468895, https://openalex.org/W2983785293, https://openalex.org/W6772532777, https://openalex.org/W6608645817, https://openalex.org/W6675931987, https://openalex.org/W6734584916, https://openalex.org/W6683497556, https://openalex.org/W2055979621, https://openalex.org/W2767609493, https://openalex.org/W3034227571, https://openalex.org/W2052902356, https://openalex.org/W1996195522, https://openalex.org/W2891638642, https://openalex.org/W2528305538, https://openalex.org/W2787302432, https://openalex.org/W3206773754, https://openalex.org/W7046297837, https://openalex.org/W2981603589, https://openalex.org/W4210909794, https://openalex.org/W4205354563, https://openalex.org/W3189470799, https://openalex.org/W2105280352, https://openalex.org/W4249205738, https://openalex.org/W2964210434, https://openalex.org/W2022164193, https://openalex.org/W4231183558, https://openalex.org/W1484867920, https://openalex.org/W2963966702, https://openalex.org/W3041129870, https://openalex.org/W4231147659, https://openalex.org/W4226516304, https://openalex.org/W2998339876, https://openalex.org/W3204236384, https://openalex.org/W597932189, https://openalex.org/W3163842339, https://openalex.org/W3194828218, https://openalex.org/W1990261069, https://openalex.org/W3099849883, https://openalex.org/W2025454433, https://openalex.org/W2594616446, https://openalex.org/W2047278710, https://openalex.org/W4288317010, https://openalex.org/W3136027957, https://openalex.org/W4286382027, https://openalex.org/W3000127803, https://openalex.org/W2902125520, https://openalex.org/W2963353224, https://openalex.org/W2158581396, https://openalex.org/W4294576319, https://openalex.org/W4288027470 |
| referenced_works_count | 81 |
| abstract_inverted_index.a | 36, 61, 90, 109, 118, 138 |
| abstract_inverted_index.We | 0 |
| abstract_inverted_index.as | 89, 135, 137 |
| abstract_inverted_index.be | 115 |
| abstract_inverted_index.by | 144 |
| abstract_inverted_index.in | 74, 81, 100, 112 |
| abstract_inverted_index.is | 35, 58, 72, 92 |
| abstract_inverted_index.of | 6, 9, 24, 27, 63, 76, 84, 129, 155, 161 |
| abstract_inverted_index.or | 44 |
| abstract_inverted_index.to | 94 |
| abstract_inverted_index.we | 50, 124 |
| abstract_inverted_index.(1) | 21 |
| abstract_inverted_index.(2) | 54 |
| abstract_inverted_index.(3) | 79 |
| abstract_inverted_index.(4) | 108 |
| abstract_inverted_index.(5) | 123 |
| abstract_inverted_index.Our | 148 |
| abstract_inverted_index.The | 17 |
| abstract_inverted_index.and | 4, 12, 31, 49, 122, 158, 169 |
| abstract_inverted_index.can | 114 |
| abstract_inverted_index.for | 39, 133, 166 |
| abstract_inverted_index.the | 2, 22, 40, 52, 55, 69, 82, 126, 151, 156 |
| abstract_inverted_index.(LP) | 30 |
| abstract_inverted_index.(QP) | 34 |
| abstract_inverted_index.are: | 20 |
| abstract_inverted_index.data | 77 |
| abstract_inverted_index.deep | 85 |
| abstract_inverted_index.form | 83 |
| abstract_inverted_index.main | 18 |
| abstract_inverted_index.some | 45, 96 |
| abstract_inverted_index.tame | 146 |
| abstract_inverted_index.that | 107 |
| abstract_inverted_index.used | 99 |
| abstract_inverted_index.well | 136 |
| abstract_inverted_index.with | 87, 117, 164 |
| abstract_inverted_index.about | 68 |
| abstract_inverted_index.basic | 97 |
| abstract_inverted_index.class | 23, 43 |
| abstract_inverted_index.layer | 91 |
| abstract_inverted_index.model | 42 |
| abstract_inverted_index.power | 57 |
| abstract_inverted_index.shown | 93 |
| abstract_inverted_index.study | 1 |
| abstract_inverted_index.terms | 75 |
| abstract_inverted_index.their | 13 |
| abstract_inverted_index.where | 66 |
| abstract_inverted_index.which | 105 |
| abstract_inverted_index.LP/QP, | 134 |
| abstract_inverted_index.bounds | 128 |
| abstract_inverted_index.convex | 10 |
| abstract_inverted_index.design | 168 |
| abstract_inverted_index.error, | 65, 104 |
| abstract_inverted_index.linear | 28 |
| abstract_inverted_index.smooth | 41 |
| abstract_inverted_index.space, | 48 |
| abstract_inverted_index.target | 70 |
| abstract_inverted_index.**first | 152 |
| abstract_inverted_index.Sobolev | 47 |
| abstract_inverted_index.discuss | 125 |
| abstract_inverted_index.generic | 139 |
| abstract_inverted_index.implies | 106 |
| abstract_inverted_index.network | 120 |
| abstract_inverted_index.numbers | 132 |
| abstract_inverted_index.problem | 141 |
| abstract_inverted_index.provide | 150 |
| abstract_inverted_index.results | 19, 149 |
| abstract_inverted_index.through | 60 |
| abstract_inverted_index.without | 103 |
| abstract_inverted_index.achieved | 116 |
| abstract_inverted_index.analysis | 102, 154 |
| abstract_inverted_index.covering | 131 |
| abstract_inverted_index.function | 71 |
| abstract_inverted_index.provided | 73 |
| abstract_inverted_index.rigorous | 153 |
| abstract_inverted_index.solution | 7, 25, 162 |
| abstract_inverted_index.(possibly | 142 |
| abstract_inverted_index.empirical | 130 |
| abstract_inverted_index.functions | 8, 26, 98 |
| abstract_inverted_index.geometry. | 147 |
| abstract_inverted_index.numerical | 101 |
| abstract_inverted_index.quadratic | 32 |
| abstract_inverted_index.reduction | 111 |
| abstract_inverted_index.universal | 37, 119 |
| abstract_inverted_index.viewpoint | 62 |
| abstract_inverted_index.exploiting | 145 |
| abstract_inverted_index.extension. | 16 |
| abstract_inverted_index.nonconvex) | 143 |
| abstract_inverted_index.properties | 160 |
| abstract_inverted_index.regression | 64 |
| abstract_inverted_index.restricted | 46 |
| abstract_inverted_index.algorithmic | 167 |
| abstract_inverted_index.approximant | 38 |
| abstract_inverted_index.functions** | 163 |
| abstract_inverted_index.guarantees. | 171 |
| abstract_inverted_index.information | 67 |
| abstract_inverted_index.multi-layer | 14 |
| abstract_inverted_index.performance | 170 |
| abstract_inverted_index.programming | 29, 33 |
| abstract_inverted_index.reconstruct | 95 |
| abstract_inverted_index.statistical | 127 |
| abstract_inverted_index.substantial | 110 |
| abstract_inverted_index.architecture | 86 |
| abstract_inverted_index.characterize | 51 |
| abstract_inverted_index.implications | 165 |
| abstract_inverted_index.investigated | 59 |
| abstract_inverted_index.learnability | 5 |
| abstract_inverted_index.optimization | 11, 88, 140 |
| abstract_inverted_index.approximation | 56, 157 |
| abstract_inverted_index.architectural | 15 |
| abstract_inverted_index.architecture, | 121 |
| abstract_inverted_index.observations, | 78 |
| abstract_inverted_index.expressibility | 3 |
| abstract_inverted_index.rate-distortion | 113 |
| abstract_inverted_index.compositionality | 80 |
| abstract_inverted_index.rate-distortion, | 53 |
| abstract_inverted_index.learning-theoretic | 159 |
| cited_by_percentile_year.max | 97 |
| cited_by_percentile_year.min | 94 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 5 |
| citation_normalized_percentile.value | 0.94607843 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |