On the minimal algebraic complexity of the rank-one approximation problem for general inner products
We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a funct...
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| creator | Kozhasov, Khazhgali Muniz, Alan Qi, Yang Sodomaco, Luca |
| description | We study the algebraic complexity of Euclidean distance minimization from a
generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED)
degree of the Segre-Veronese variety counts the number of complex critical
points of this optimization problem. We regard this invariant as a function of
inner products. We prove that Frobenius inner product is a local minimum of the
ED degree, and conjecture that it is a global minimum. We prove our conjecture
in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors.
We discuss the above optimization problem for other algebraic varieties,
classifying all possible values of the ED degree. Our approach combines tools
from Singularity Theory, Morse Theory, and Algebraic Geometry. |
| doi_str_mv | 10.48550/arxiv.2309.15105 |
| format | Article |
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generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED)
degree of the Segre-Veronese variety counts the number of complex critical
points of this optimization problem. We regard this invariant as a function of
inner products. We prove that Frobenius inner product is a local minimum of the
ED degree, and conjecture that it is a global minimum. We prove our conjecture
in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors.
We discuss the above optimization problem for other algebraic varieties,
classifying all possible values of the ED degree. Our approach combines tools
from Singularity Theory, Morse Theory, and Algebraic Geometry.</description><identifier>DOI: 10.48550/arxiv.2309.15105</identifier><language>eng</language><subject>Mathematics - Algebraic Geometry ; Mathematics - Optimization and Control</subject><creationdate>2023-09</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>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2309.15105$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2309.15105$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Kozhasov, Khazhgali</creatorcontrib><creatorcontrib>Muniz, Alan</creatorcontrib><creatorcontrib>Qi, Yang</creatorcontrib><creatorcontrib>Sodomaco, Luca</creatorcontrib><title>On the minimal algebraic complexity of the rank-one approximation problem for general inner products</title><description>We study the algebraic complexity of Euclidean distance minimization from a
generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED)
degree of the Segre-Veronese variety counts the number of complex critical
points of this optimization problem. We regard this invariant as a function of
inner products. We prove that Frobenius inner product is a local minimum of the
ED degree, and conjecture that it is a global minimum. We prove our conjecture
in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors.
We discuss the above optimization problem for other algebraic varieties,
classifying all possible values of the ED degree. Our approach combines tools
from Singularity Theory, Morse Theory, and Algebraic Geometry.</description><subject>Mathematics - Algebraic Geometry</subject><subject>Mathematics - Optimization and Control</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotj8tqwzAURLXpoqT9gK6qH7Ar6eqRLEPoCwLZZG-uHk5FbMkobnH-vo7b1QzMMMwh5ImzWq6VYi9YpvhTC2CbmivO1D3xh0THr0D7mGKPHcXuFGzB6KjL_dCFKY5XmtulUzCdq5wCxWEoeZr7Y8yJzt52oadtLvQUUijzTEyz3hL_7cbLA7lrsbuEx39dkePb63H3Ue0P75-77b5CbVQFaLj3rQqaWy01NwIdEyg9SuYMZ9p7BtLDZi1Aa7DOt5wDCOEBjLIBVuT5b3bBbIYyPyzX5obbLLjwC7nhUSw</recordid><startdate>20230926</startdate><enddate>20230926</enddate><creator>Kozhasov, Khazhgali</creator><creator>Muniz, Alan</creator><creator>Qi, Yang</creator><creator>Sodomaco, Luca</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20230926</creationdate><title>On the minimal algebraic complexity of the rank-one approximation problem for general inner products</title><author>Kozhasov, Khazhgali ; Muniz, Alan ; Qi, Yang ; Sodomaco, Luca</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a675-3a71ddf5e61b646172ac02a4da40c7106dd034d39823663bcdf113322d3375be3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics - Algebraic Geometry</topic><topic>Mathematics - Optimization and Control</topic><toplevel>online_resources</toplevel><creatorcontrib>Kozhasov, Khazhgali</creatorcontrib><creatorcontrib>Muniz, Alan</creatorcontrib><creatorcontrib>Qi, Yang</creatorcontrib><creatorcontrib>Sodomaco, Luca</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Kozhasov, Khazhgali</au><au>Muniz, Alan</au><au>Qi, Yang</au><au>Sodomaco, Luca</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the minimal algebraic complexity of the rank-one approximation problem for general inner products</atitle><date>2023-09-26</date><risdate>2023</risdate><abstract>We study the algebraic complexity of Euclidean distance minimization from a
generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED)
degree of the Segre-Veronese variety counts the number of complex critical
points of this optimization problem. We regard this invariant as a function of
inner products. We prove that Frobenius inner product is a local minimum of the
ED degree, and conjecture that it is a global minimum. We prove our conjecture
in the case of matrices and symmetric binary and $3\times 3\times 3$ tensors.
We discuss the above optimization problem for other algebraic varieties,
classifying all possible values of the ED degree. Our approach combines tools
from Singularity Theory, Morse Theory, and Algebraic Geometry.</abstract><doi>10.48550/arxiv.2309.15105</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Algebraic Geometry Mathematics - Optimization and Control |
| title | On the minimal algebraic complexity of the rank-one approximation problem for general inner products |
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