Total variation regularized Fréchet regression for metric-space valued data

IF 3.2 1区 数学 Q1 STATISTICS & PROBABILITY
Zhenhua Lin,Hans-Georg Müller
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引用次数: 13

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 Frechet regression, which refers to a regression setting where a response residing in a generic metric space is paired with a scalar predictor and the target is a conditional Frechet mean. Specifically, we seek to approximate an unknown metric-space valued function by an estimator that minimizes the Frechet 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.
度量空间值数据的总变分正则化fr回归
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来源期刊
Annals of Statistics
Annals of Statistics 数学-统计学与概率论
CiteScore
9.30
自引率
8.90%
发文量
119
审稿时长
6-12 weeks
期刊介绍: The Annals of Statistics aim to publish research papers of highest quality reflecting the many facets of contemporary statistics. Primary emphasis is placed on importance and originality, not on formalism. The journal aims to cover all areas of statistics, especially mathematical statistics and applied & interdisciplinary statistics. Of course many of the best papers will touch on more than one of these general areas, because the discipline of statistics has deep roots in mathematics, and in substantive scientific fields.
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