Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method
In this work, we propose a novel implementation of Newton's method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of $p$-norms, $1
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| creator | Dieci, Luca Omarov, Daniyar |
| description | In this work, we propose a novel implementation of Newton's method for
solving semi-discrete optimal transport (OT) problems for cost functions which
are a positive combination of $p$-norms, $1 |
| doi_str_mv | 10.48550/arxiv.2310.07489 |
| format | Article |
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solving semi-discrete optimal transport (OT) problems for cost functions which
are a positive combination of $p$-norms, $1<p<\infty$. It is well understood
that the solution of a semi-discrete OT problem is equivalent to finding a
partition of a bounded region in Laguerre cells, and we prove that the Laguerre
cells are star-shaped with respect to the target points. By exploiting the
geometry of the Laguerre cells, we obtain an efficient and reliable
implementation of Newton's method to find the sought network structure. We
provide implementation details and extensive results in support of our
technique in 2-d problems, as well as comparison with other approaches used in
the literature.</description><identifier>DOI: 10.48550/arxiv.2310.07489</identifier><language>eng</language><subject>Computer Science - Numerical Analysis ; Mathematics - Numerical Analysis ; Mathematics - Optimization and Control</subject><creationdate>2023-10</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2310.07489$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2310.07489$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Dieci, Luca</creatorcontrib><creatorcontrib>Omarov, Daniyar</creatorcontrib><title>Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method</title><description>In this work, we propose a novel implementation of Newton's method for
solving semi-discrete optimal transport (OT) problems for cost functions which
are a positive combination of $p$-norms, $1<p<\infty$. It is well understood
that the solution of a semi-discrete OT problem is equivalent to finding a
partition of a bounded region in Laguerre cells, and we prove that the Laguerre
cells are star-shaped with respect to the target points. By exploiting the
geometry of the Laguerre cells, we obtain an efficient and reliable
implementation of Newton's method to find the sought network structure. We
provide implementation details and extensive results in support of our
technique in 2-d problems, as well as comparison with other approaches used in
the literature.</description><subject>Computer Science - Numerical Analysis</subject><subject>Mathematics - Numerical Analysis</subject><subject>Mathematics - Optimization and Control</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotjztPwzAUhb0woMIPYMIbU4oTOw-zofKqVNFKjbpG17k3NFJiR7ZV4N8TCtM5OsOn8zF2k4qlqvJc3IP_6k_LTM6DKFWlL9lh74ZTbz_4nsY-eepD6ykS306xH2HgtQcbJucj33lnBhrDAw8RPA9HmAjJUggcLPJ3-ozO3gU-Ujw6vGIXHQyBrv9zweqX53r1lmy2r-vV4yaBotSJKjthKjI5KlFmmGKGRSdQktGyAKW1QpkVRd6CnAsBImrE1qQVEmW5kQt2-4c9izWTn0_77-ZXsDkLyh_B800r</recordid><startdate>20231011</startdate><enddate>20231011</enddate><creator>Dieci, Luca</creator><creator>Omarov, Daniyar</creator><scope>AKY</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20231011</creationdate><title>Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method</title><author>Dieci, Luca ; Omarov, Daniyar</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a679-47f0b8eb5d4072d1d2d6f0d3eb936a4994d32665ca3d32eaddd9ddcb18dee25b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Computer Science - Numerical Analysis</topic><topic>Mathematics - Numerical Analysis</topic><topic>Mathematics - Optimization and Control</topic><toplevel>online_resources</toplevel><creatorcontrib>Dieci, Luca</creatorcontrib><creatorcontrib>Omarov, Daniyar</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Dieci, Luca</au><au>Omarov, Daniyar</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method</atitle><date>2023-10-11</date><risdate>2023</risdate><abstract>In this work, we propose a novel implementation of Newton's method for
solving semi-discrete optimal transport (OT) problems for cost functions which
are a positive combination of $p$-norms, $1<p<\infty$. It is well understood
that the solution of a semi-discrete OT problem is equivalent to finding a
partition of a bounded region in Laguerre cells, and we prove that the Laguerre
cells are star-shaped with respect to the target points. By exploiting the
geometry of the Laguerre cells, we obtain an efficient and reliable
implementation of Newton's method to find the sought network structure. We
provide implementation details and extensive results in support of our
technique in 2-d problems, as well as comparison with other approaches used in
the literature.</abstract><doi>10.48550/arxiv.2310.07489</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Computer Science - Numerical Analysis Mathematics - Numerical Analysis Mathematics - Optimization and Control |
| title | Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method |
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