Abstract
Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparametric Fréchet regression, which refers to a regression setting where a response residing in a metric space is paired with a scalar predictor and the target is a conditional Fréchet mean. Specifically, we seek to approximate an unknown metric-space valued function by an estimator that minimizes the Fréchet version of least squares and at the same time has small total variation, appropriately defined for metric-space valued objects. We show that the resulting estimator is representable by a piece-wise constant function and establish the minimax convergence rate of the proposed estimator for metric data objects that reside in Hadamard spaces. We illustrate the numerical performance of the proposed method for both simulated and real data, including metric spaces of symmetric positive-definite matrices with the affine-invariant distance, of probability distributions on the real line with the Wasserstein distance, and of phylogenetic trees with the Billera–Holmes–Vogtmann metric.
Funding Statement
The first author was supported by NUS Startup Grant R-155-000-217-133.
The second author was supported by NSF Grants DMS-1712864 and DMS-2014626.
Acknowledgments
We extend our sincere thanks to the Editor, Associate Editor and several referees for their constructive comments that lead to numerous improvements over a previous version. Data were provided in part by the Human Connectome Project, WU-Minn Consortium (PI: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
Funding Statement
The first author was supported by NUS Startup Grant R-155-000-217-133.
The second author was supported by NSF Grants DMS-1712864 and DMS-2014626.
Acknowledgments
We extend our sincere thanks to the Editor, Associate Editor and several referees for their constructive comments that lead to numerous improvements over a previous version. Data were provided in part by the Human Connectome Project, WU-Minn Consortium (PI: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
Citation
Zhenhua Lin. Hans-Georg Müller. "Total variation regularized Fréchet regression for metric-space valued data." Ann. Statist. 49 (6) 3510 - 3533, December 2021. https://doi.org/10.1214/21-AOS2095
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