Projections of Tropical Fermat-Weber Points
This paper is motivated by the difference between the classical principal component analysis (PCA) in a Euclidean space and the tropical PCA in a tropical projective torus as follows. In Euclidean space, the projection of the mean point of a given data set on the principle component is the mean poin...
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MDPI AG
2021-12-01
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| author | Weiyi Ding Xiaoxian Tang |
| author_facet | Weiyi Ding Xiaoxian Tang |
| author_sort | Weiyi Ding |
| collection | DOAJ |
| description | This paper is motivated by the difference between the classical principal component analysis (PCA) in a Euclidean space and the tropical PCA in a tropical projective torus as follows. In Euclidean space, the projection of the mean point of a given data set on the principle component is the mean point of the projection of the data set. However, in tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set on a tropical polytope is a Fermat-Weber point of the projection of the data set. This is caused by the difference between the Euclidean metric and the tropical metric. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm and its improved version, such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">R</mi></semantics></math></inline-formula> language and test how they work with random data sets. We also use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">R</mi></semantics></math></inline-formula> language for numerical computation. The experimental results show that these algorithms are stable and efficient, with a high success rate. |
| first_indexed | 2024-03-10T04:48:14Z |
| format | Article |
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| language | English |
| last_indexed | 2024-03-10T04:48:14Z |
| publishDate | 2021-12-01 |
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| spelling | doaj.art-44c6ef51a93b47699f4666b1664088722023-11-23T02:46:05ZengMDPI AGMathematics2227-73902021-12-01923310210.3390/math9233102Projections of Tropical Fermat-Weber PointsWeiyi Ding0Xiaoxian Tang1School of Mathematical Sciences, Beihang University, Beijing 100191, ChinaSchool of Mathematical Sciences, Beihang University, Beijing 100191, ChinaThis paper is motivated by the difference between the classical principal component analysis (PCA) in a Euclidean space and the tropical PCA in a tropical projective torus as follows. In Euclidean space, the projection of the mean point of a given data set on the principle component is the mean point of the projection of the data set. However, in tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set on a tropical polytope is a Fermat-Weber point of the projection of the data set. This is caused by the difference between the Euclidean metric and the tropical metric. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm and its improved version, such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">R</mi></semantics></math></inline-formula> language and test how they work with random data sets. We also use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">R</mi></semantics></math></inline-formula> language for numerical computation. The experimental results show that these algorithms are stable and efficient, with a high success rate.https://www.mdpi.com/2227-7390/9/23/3102Fermat-Weber pointconvex polytopetropical projectiontropical PCA |
| spellingShingle | Weiyi Ding Xiaoxian Tang Projections of Tropical Fermat-Weber Points Mathematics Fermat-Weber point convex polytope tropical projection tropical PCA |
| title | Projections of Tropical Fermat-Weber Points |
| title_full | Projections of Tropical Fermat-Weber Points |
| title_fullStr | Projections of Tropical Fermat-Weber Points |
| title_full_unstemmed | Projections of Tropical Fermat-Weber Points |
| title_short | Projections of Tropical Fermat-Weber Points |
| title_sort | projections of tropical fermat weber points |
| topic | Fermat-Weber point convex polytope tropical projection tropical PCA |
| url | https://www.mdpi.com/2227-7390/9/23/3102 |
| work_keys_str_mv | AT weiyiding projectionsoftropicalfermatweberpoints AT xiaoxiantang projectionsoftropicalfermatweberpoints |