  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.
