Supervised Homogeneity Fusion: a Combinatorial Approach
This work addresses the challenge of parameter dimension reduction in regression for statisticians and data scientists, offering a novel method with theoretical guarantees, though it is incremental in the context of existing fusion techniques.
The paper tackles the problem of fusing regression coefficients into homogeneous groups to improve statistical accuracy, proposing a combinatorial approach called L0-Fusion that achieves grouping consistency under minimal conditions and shows superiority in grouping accuracy in simulations and real data.
Fusing regression coefficients into homogenous 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 $L_0$-Fusion that is amenable to mixed integer optimization (MIO). On the statistical aspect, we identify a fundamental quantity called grouping sensitivity that underpins the difficulty of recovering the true groups. We show that $L_0$-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 $L_0$-Fusion coupled 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 a MIO formulation for $L_0$-Fusion along with a warm start strategy. Simulation and real data analysis demonstrate that $L_0$-Fusion exhibits superiority over its competitors in terms of grouping accuracy.