February 2024 Supervised homogeneity fusion: A combinatorial approach
Wen Wang, Shihao Wu, Ziwei Zhu, Ling Zhou, Peter X.-K. Song
Author Affiliations +
Ann. Statist. 52(1): 285-310 (February 2024). DOI: 10.1214/23-AOS2347


Fusing regression coefficients into homogeneous groups can unveil those coefficients that share a common value within each group. Such groupwise homogeneity reduces the intrinsic dimension of the parameter space and unleashes sharper statistical accuracy. We propose and investigate a new combinatorial grouping approach called L0-Fusion that is amenable to mixed integer optimization (MIO). On the statistical aspect, we identify a fundamental quantity called MSE grouping sensitivity that underpins the difficulty of recovering the true groups. We show that L0-Fusion achieves grouping consistency under the weakest possible requirement of the grouping sensitivity: if this requirement is violated, then the minimax risk of group misspecification will fail to converge to zero. Moreover, we show that in the high-dimensional regime, one can apply L0-Fusion with a sure screening set of features without any essential loss of statistical efficiency, while reducing the computational cost substantially. On the algorithmic aspect, we provide an MIO formulation for L0-Fusion along with a warm start strategy. Simulation and real data analysis demonstrate that L0-Fusion exhibits superiority over its competitors in terms of grouping accuracy.

Funding Statement

Dr. Song’s research is supported by National Institutes of Health grants R01ES024732 and R01ES033656 and National Science Foundation grants DMS2113564 and DMS2015366.
Dr. Zhou’s research was partially supported by the National Natural Science Foundation of China (Nos. 12271441 and 11931014).


We thank the Co-editors, the Associate Editor and two anonymous reviewers for constructive comments that have led to a significantly improved manuscript. We also thank Drs. Z.T. Ke and X. Shen for providing their Matlab code and R code for CARDS and FGSG methods, respectively. Wang and Wu are co-first authors of this paper.


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Wen Wang. Shihao Wu. Ziwei Zhu. Ling Zhou. Peter X.-K. Song. "Supervised homogeneity fusion: A combinatorial approach." Ann. Statist. 52 (1) 285 - 310, February 2024. https://doi.org/10.1214/23-AOS2347


Received: 1 November 2022; Revised: 1 October 2023; Published: February 2024
First available in Project Euclid: 7 March 2024

MathSciNet: MR4718416
Digital Object Identifier: 10.1214/23-AOS2347

Primary: 60J05 , 90C11
Secondary: 62P10

Keywords: clustering , Dimension reduction , linear model , mixed integer optimization , supervised learning

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 1 • February 2024
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