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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2411.10889 |
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| _version_ | 1866912171159453696 |
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| author | Deng, Chengyuan Gao, Jie Lu, Kevin Luo, Feng Sun, Hongbin Xin, Cheng |
| author_facet | Deng, Chengyuan Gao, Jie Lu, Kevin Luo, Feng Sun, Hongbin Xin, Cheng |
| contents | We introduce Non-Euclidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10889 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms Deng, Chengyuan Gao, Jie Lu, Kevin Luo, Feng Sun, Hongbin Xin, Cheng Machine Learning We introduce Non-Euclidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods. |
| title | Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2411.10889 |