.png){kind=icon}
返回
Deep Fréchet Regression
.png){kind=icon}
PRE
AI
DOI：10.1080/01621459.2025.2507982.png)
.png){kind=icon}
原文求助.png){kind=icon}
分享.png){kind=icon}
收藏
摘要
En 中文
Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This article addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean predictors. We propose a flexible regression model capable of handling high-dimensional predictors without imposing parametric assumptions. Two primary challenges are addressed: the curse of dimensionality in nonparametric regression and the absence of linear structure in general metric spaces. The former is tackled using deep neural networks, while for the latter we demonstrate the feasibility of mapping the metric space where responses reside to a low-dimensional Euclidean space using manifold learning. We introduce a reverse mapping approach, employing local Fréchet regression, to map the low-dimensional manifold representations back to objects in the original metric space. We develop a theoretical framework, investigating the convergence rate of deep neural networks under dependent sub-Gaussian noise with bias. The convergence rate of the proposed regression model is then obtained by expanding the scope of local Fréchet regression to accommodate multivariate predictors in the presence of errors in predictors. Simulations and case studies show that the proposed model outperforms existing methods for non-Euclidean responses, focusing on the special cases of probability distributions and networks. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.
Keyword:
Curse of dimensionality
Deep learning
Fréchet regression
Non-Euclidean data
Wasserstein space
期刊
J
IF:
3
论文数:
5.2K
被引数:
4.8W
机构
引用论文
Sturm, K.-T. (2003), “Probability Measures on Metric Spaces of Nonpositive Curvature,” in Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Contemporary Mathematics, 338, pp. 357–390. Providence, RI: American Mathematical Society. (Open in a new window)Google ScholarSturm, K.-T. (2003), 《负曲率度量空间上的概率测度》, 载于《流形、图和度量空间上的热核与分析》(巴黎, 2002), Contemporary Mathematics, 338, 第357-390页。Providence, RI: American Mathematical Society. (在新窗口中打开)Google Scholar.png)
IF0
PREAI
分享收藏Ying, C., and Yu, Z. (2022), “Fréchet Sufficient Dimension Reduction for Random Objects,” Biometrika, 109, 975–992. DOI: 10.1093/biomet/asac012. (Open in a new window)Web of Science ®(Open in a new window)Google ScholarYing, C., and Yu, Z. (2022), “Fréchet充分降维方法用于随机对象,” Biometrika, 109, 975–992. DOI: 10.1093/biomet/asac012. (在新窗口中打开) Web of Science ®(在新窗口中打开) Google Scholar
IF0
PREAI
分享收藏Zhang, C., Kokoszka, P., and Petersen, A. (2022), “Wasserstein Autoregressive Models for Density Time Series,” Journal of Time Series Analysis, 43, 30–52. DOI: 10.1111/jtsa.12590. (Open in a new window)Web of Science ®(Open in a new window)Google Scholar张, C., Kokoszka, P., and Petersen, A. (2022), “Wasserstein Autoregressive Models for Density Time Series,” Journal of Time Series Analysis, 43, 30–52. DOI: 10.1111/jtsa.12590. (在新窗口中打开)Web of Science ®(在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏Anthony, M., Bartlett, P. L., and Bartlett, P. L. (1999), Neural Network Learning: Theoretical Foundations (Vol. 9), Cambridge: Cambridge University Press. (Open in a new window)Google ScholarAnthony, M., Bartlett, P. L., and Bartlett, P. L. (1999), Neural Network Learning: Theoretical Foundations (Vol. 9), Cambridge: Cambridge University Press. (在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏Faraway, J. J. (2014), “Regression for Non-Euclidean Data Using Distance Matrices,” Journal of Applied Statistics, 41, 2342–2357. DOI: 10.1080/02664763.2014.909794. (Open in a new window)Web of Science ®(Open in a new window)Google ScholarFaraway, J. J. (2014), “使用距离矩阵对非欧几里得数据进行回归分析,” 《应用统计学杂志》, 41, 2342–2357. DOI: 10.1080/02664763.2014.909794. (在新窗口中打开) Web of Science ®(在新窗口中打开) Google Scholar
IF0
PREAI
分享收藏Nye, T. M., Tang, X., Weyenberg, G., and Yoshida, R. (2017), “Principal Component Analysis and the Locus of the Fréchet Mean in the Space of Phylogenetic Trees,” Biometrika, 104, 901–922. DOI: 10.1093/biomet/asx047. (Open in a new window)PubMed(Open in a new window)Google ScholarNye, T. M., Tang, X., Weyenberg, G., and Yoshida, R. (2017), “主成分分析及系统发育树空间中Fréchet均值的分布位置,” Biometrika, 104, 901–922. DOI: 10.1093/biomet/asx047. (在新窗口中打开)PubMed(在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏Van der Vaart, A., and Wellner, J. (2023), Weak Convergence and Empirical Processes: With Applications to Statistics, New York: Springer. (Open in a new window)Google ScholarVan der Vaart, A., and Wellner, J. (2023), 《弱收敛与经验过程：统计应用》，纽约：施普林格出版社。 (在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏Coifman, R. R., and Lafon, S. (2006), “Diffusion Maps,” Applied and Computational Harmonic Analysis, 21, 5–30. DOI: 10.1016/j.acha.2006.04.006. (Open in a new window)Web of Science ®(Open in a new window)Google ScholarCoifman, R. R., 和 Lafon, S. (2006), “Diffusion Maps,” 应用与计算谐波分析, 21, 5–30. DOI: 10.1016/j.acha.2006.04.006. (在新窗口中打开) Web of Science ®(在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏Delicado, P. (2011), “Dimensionality Reduction When Data are Density Functions,” Computational Statistics & Data Analysis, 55, 401–420. DOI: 10.1016/j.csda.2010.05.008. (Open in a new window)Web of Science ®(Open in a new window)Google ScholarDelicado, P. (2011), “Dimensionality Reduction When Data are Density Functions,” Computational Statistics & Data Analysis, 55, 401–420. DOI: 10.1016/j.csda.2010.05.008. (在新窗口中打开)Web of Science ®(在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏Kurisu, D., Fukami, R., and Koike, Y. (2025), “Adaptive Deep Learning for Nonparametric Time Series Regression,” Bernoulli, 31, 240–270. (Open in a new window)Google ScholarKurisu, D., Fukami, R., and Koike, Y. (2025), “Adaptive Deep Learning for Nonparametric Time Series Regression,” Bernoulli, 31, 240–270. (在新窗口中打开)Google Scholar
IF0
PREAI
分享收藏
查看更多内容