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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2205.15059 |
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| _version_ | 1866911771594326016 |
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| author | Li, Tao Meng, Cheng Xu, Hongteng Yu, Jun |
| author_facet | Li, Tao Meng, Cheng Xu, Hongteng Yu, Jun |
| contents | Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the $L_p$ cost in the $d$-dimensional space converges to its population counterpart at a rate of no more than $O(n^{-1/2\max\{d,p\}})$. To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_15059 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hilbert Curve Projection Distance for Distribution Comparison Li, Tao Meng, Cheng Xu, Hongteng Yu, Jun Machine Learning Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the $L_p$ cost in the $d$-dimensional space converges to its population counterpart at a rate of no more than $O(n^{-1/2\max\{d,p\}})$. To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance. |
| title | Hilbert Curve Projection Distance for Distribution Comparison |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2205.15059 |